1. The document discusses methods for solving systems of linear equations and calculating eigen values and eigen vectors of matrices. It describes direct and iterative methods for solving linear systems, including Gauss-Jacobi and Gauss-Seidel iterative methods.
2. It also covers the concepts of diagonal dominance and consistency conditions for linear systems. Rayleigh's power method is introduced for finding the dominant eigen value and vector of a matrix.
3. Examples are provided to illustrate solving linear systems by Jacobi's method and checking for diagonal dominance and consistency of systems. The convergence criteria for Gauss-Jacobi and Gauss-Seidel methods are also outlined.
This document discusses arithmetic sequences. It defines key terms like sequence, term, and common difference. It explains how to identify if a set of numbers forms an arithmetic sequence based on having a constant difference between terms. Methods are provided for finding the next term, the nth term, and representing the sequence as a linear function. Examples demonstrate how to apply these concepts to solve problems involving arithmetic sequences.
1) The document defines and provides examples of different types of angles including adjacent angles, complementary angles, supplementary angles, vertical angles, and angles formed when parallel lines are cut by a transversal.
2) It then provides practice problems involving calculating missing angles using properties such as angles summing to 90, 180 degrees, and the interior angles of triangles summing to 180 degrees.
3) The final problems involve identifying congruent, supplementary, and corresponding angles related to two parallel lines cut by a transversal.
The three undefined terms in geometry are point, line, and plane. A point indicates a position in space and is named with capital letters or coordinates. A line is an infinite set of adjacent points that extends in both directions, named using two points or a lowercase letter. A plane is a flat surface that extends indefinitely in all directions, named using three points or an uppercase letter.
Unit 9 lesson 6 polygons in the coordinate planemlabuski
The document provides worksheets and homework problems for students to practice calculating areas and perimeters of polygons in the coordinate plane. Students are given polygons with vertices defined by coordinate pairs and are asked to graph the polygons, identify their shape, and calculate their areas. Some polygons are comprised of multiple shapes requiring the use of area formulas to calculate the overall area.
This document provides an overview of axiomatic systems and their components. It begins by defining the key terms in an axiomatic system, including undefined terms, defined terms, axioms, and theorems. Examples are given of simple axiomatic systems like four point geometry. Properties of axiomatic systems are also discussed, such as consistency, independence, completeness, and duality. Finally, finite projective planes are introduced as an important type of finite geometry that satisfies additional axioms beyond basic systems.
This document provides definitions and concepts related to analytic geometry. It discusses the Cartesian coordinate system, ordered pairs, axes, and coordinates. It defines distance formulas for horizontal, vertical, and slant line segments. Sample problems are provided to calculate distances between points and to determine geometric properties related to triangles and rectangles on a coordinate plane. The objectives are to familiarize students with the coordinate system and to determine distances, slopes, angles of inclination for lines and line segments.
The document discusses basic concepts in geometry including points, lines, planes, and their relationships. It defines a point as having no size or shape, a line as connecting two or more points and extending indefinitely in both directions, and a plane as a flat two-dimensional surface containing points and lines. The document provides examples of naming points, lines, and planes and identifies collinear points that lie on the same line and coplanar points that lie on the same plane. It includes practice problems asking students to name, draw, and identify various geometric concepts.
More Free Resources to Help You Teach your Geometry Lesson on Points Lines and Planes can be found here:
https://geometrycoach.com/points-lines-and-planes/
If you are looking for more great lesson ideas sign up for our FREEBIES at:
Pre Algebra: https://prealgebracoach.com/unit
Algebra 1: https://algebra1coach.com/unit
Geometry: https://geometrycoach.com/optin
Algebra 2 with Trigonometry: https://algebra2coach.com/unit
This document discusses arithmetic sequences. It defines key terms like sequence, term, and common difference. It explains how to identify if a set of numbers forms an arithmetic sequence based on having a constant difference between terms. Methods are provided for finding the next term, the nth term, and representing the sequence as a linear function. Examples demonstrate how to apply these concepts to solve problems involving arithmetic sequences.
1) The document defines and provides examples of different types of angles including adjacent angles, complementary angles, supplementary angles, vertical angles, and angles formed when parallel lines are cut by a transversal.
2) It then provides practice problems involving calculating missing angles using properties such as angles summing to 90, 180 degrees, and the interior angles of triangles summing to 180 degrees.
3) The final problems involve identifying congruent, supplementary, and corresponding angles related to two parallel lines cut by a transversal.
The three undefined terms in geometry are point, line, and plane. A point indicates a position in space and is named with capital letters or coordinates. A line is an infinite set of adjacent points that extends in both directions, named using two points or a lowercase letter. A plane is a flat surface that extends indefinitely in all directions, named using three points or an uppercase letter.
Unit 9 lesson 6 polygons in the coordinate planemlabuski
The document provides worksheets and homework problems for students to practice calculating areas and perimeters of polygons in the coordinate plane. Students are given polygons with vertices defined by coordinate pairs and are asked to graph the polygons, identify their shape, and calculate their areas. Some polygons are comprised of multiple shapes requiring the use of area formulas to calculate the overall area.
This document provides an overview of axiomatic systems and their components. It begins by defining the key terms in an axiomatic system, including undefined terms, defined terms, axioms, and theorems. Examples are given of simple axiomatic systems like four point geometry. Properties of axiomatic systems are also discussed, such as consistency, independence, completeness, and duality. Finally, finite projective planes are introduced as an important type of finite geometry that satisfies additional axioms beyond basic systems.
This document provides definitions and concepts related to analytic geometry. It discusses the Cartesian coordinate system, ordered pairs, axes, and coordinates. It defines distance formulas for horizontal, vertical, and slant line segments. Sample problems are provided to calculate distances between points and to determine geometric properties related to triangles and rectangles on a coordinate plane. The objectives are to familiarize students with the coordinate system and to determine distances, slopes, angles of inclination for lines and line segments.
The document discusses basic concepts in geometry including points, lines, planes, and their relationships. It defines a point as having no size or shape, a line as connecting two or more points and extending indefinitely in both directions, and a plane as a flat two-dimensional surface containing points and lines. The document provides examples of naming points, lines, and planes and identifies collinear points that lie on the same line and coplanar points that lie on the same plane. It includes practice problems asking students to name, draw, and identify various geometric concepts.
More Free Resources to Help You Teach your Geometry Lesson on Points Lines and Planes can be found here:
https://geometrycoach.com/points-lines-and-planes/
If you are looking for more great lesson ideas sign up for our FREEBIES at:
Pre Algebra: https://prealgebracoach.com/unit
Algebra 1: https://algebra1coach.com/unit
Geometry: https://geometrycoach.com/optin
Algebra 2 with Trigonometry: https://algebra2coach.com/unit
Trigonometry deals with relationships between sides and angles in right triangles. The three main trigonometric functions are sine, cosine, and tangent, which provide ratios of sides of a right triangle based on a given angle. A scientific calculator can be used to find trigonometric ratios for any angle, as well as to calculate missing angles or sides of a right triangle when two pieces of information are known.
The document introduces some basic concepts in geometry, including:
1. Points, lines, and planes are undefined terms that form the foundations of geometry.
2. It explains concepts like collinear points, coplanar points, line segments, rays, and how to classify angles.
3. It discusses intersections of lines, planes, and examples of modeling intersections of geometric figures.
Lesson Plan Math : History of measurementxjason001
This document provides an overview of the history of measurement. It discusses how early civilizations used parts of the body to measure things, with the Egyptians using cubits and digits, the Babylonians using minas, and the Chinese using chihs and shihs. Problems arose with this non-standard system, leading to standardized systems like the modern metric system. The document also outlines some key units in other historical systems like those used by the Romans, English, and the eventual development of the International System of Units.
Points, lines, and planes are the basic building blocks of geometry. A point is a location without shape and is represented by a capital letter. A line contains points and has no thickness, with exactly one line passing through any two points. The intersection of two lines is a point. A plane is a flat surface made up of points that extends infinitely in all directions, with the intersection of two planes being a line. Planes are identified by a capital italicized letter or by three non-collinear points.
The document discusses various techniques for instrument drawing and technical lettering. It covers topics such as common drawing equipment like drawing boards, T-squares, and triangles used to draw straight lines and specific angles. It also discusses drawing pencils, lead grades, and their applications for different line types. Specific techniques are covered for drawing horizontal and vertical lines, using a compass to draw circles, and sharpening compass leads. Guidelines are provided for pencil drawing, contrasting line weights, and single-stroke Gothic lettering.
The document discusses sets of axioms and finite geometries. It begins by defining geometry as "earth measure" from its Greek roots and discusses some early examples of geometry like Eratosthenes' measurement of the circumference of the Earth. It then discusses the development of geometry through the Greeks and Euclid, including undefined terms, axioms, postulates, and theorems. It provides examples of Euclid's axioms and postulates and discusses modern refinements. Finally, it introduces finite geometries, which have a finite number of elements, and provides an example of a three-point geometry with its axioms and a proof of one of its theorems.
This video discusses the undefined geometric terms of point, line, and plane. It provides examples of objects that can represent these terms, such as the tip of a needle for a point, an electric wire for a line, and the surface of a table for a plane. The video calls these the "undefined terms" because they do not require further explanation and defines each one as having no dimension for a point, being a set of arranged points for a line, and being a flat endless surface for a plane. It includes classroom activities where students match objects to the terms and examples of how to represent and identify a point, line, and plane.
Math 7 geometry 02 postulates and theorems on points, lines, and planesGilbert Joseph Abueg
This document covers basic concepts in geometry including:
1. Definitions, undefined terms, postulates, and theorems related to points, lines, and planes. Undefined terms include points, lines, and planes. Definitions clearly define concepts like line segments.
2. Postulates are statements accepted as true without proof, including the ruler postulate, segment addition postulate, and plane postulate.
3. Theorems are important statements that can be proven, such as the intersection of lines theorem and the theorem regarding a line and point determining a unique plane.
The document provides information about triangle congruence, including:
1. There are three postulates for proving triangles are congruent: side-side-side (SSS), side-angle-side (SAS), and angle-side-angle (ASA).
2. The SSS postulate states that if three sides of one triangle are congruent to three sides of a second triangle, then the triangles are congruent.
3. The SAS postulate states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the triangles are congruent.
4. The ASA postulate states that if two angles
This document provides information on spherical trigonometry. It defines key terms like great circles, small circles, spherical angles, and spherical triangles. It describes properties of these concepts, such as every great circle passing through the center of a sphere. Formulas for solving spherical triangles are presented, including the Sine Formula, Cosine Formula, and Haversine Formula. Examples show how to use these formulas to calculate unknown sides and angles of spherical triangles given other information.
K TO 12 GRADE 7 LEARNING MODULE IN MATHEMATICS (Quarter 3)LiGhT ArOhL
This document provides a lesson plan on solving linear equations and inequalities in one variable algebraically. It begins with reviewing translating between verbal and mathematical phrases and evaluating expressions. The main focus is on introducing and applying the properties of equality, including reflexive, symmetric, transitive, and substitution properties, to solve equations algebraically. Word problems involving equations in one variable are also discussed. The objectives are to identify and apply the properties of equality to find solutions to equations and solve word problems involving one variable equations.
This document summarizes a Grade 8 mathematics lesson on triangle congruency. The objectives were for students to define and illustrate congruent triangles, identify corresponding sides and angles, and present solutions with accuracy. Content covered triangle congruence. Learning activities included visual aids, measuring triangles with meter sticks, forming pairs of congruent triangles, and applying concepts to a real-world problem of balancing swing supports. Students were evaluated on identifying corresponding parts of congruent triangles. The teacher reflected on students' mastery levels and effectiveness of instructional strategies.
The document provides instructions for organizing and presenting statistical data using frequency tables and histograms. It discusses how to construct a frequency table by grouping raw data into intervals and tallying the frequencies. It then explains how to create a histogram by using the frequency table to draw rectangles whose widths represent intervals and heights represent frequencies. The lesson emphasizes that frequency tables and histograms are useful tools for organizing large data sets and communicating patterns in the data visually.
This document provides information and examples for determining congruence and similarity using slopes of lines and triangles. It explains that the ratio of the rise to the run of two slope triangles formed by a line is equal to the slope of the line. It also gives examples of finding the slope of a roof, stairs, and verifying that the slope is the same between points on the line. Finally, it discusses that right triangles that have their hypotenuses on the same line in a coordinate plane are called slope triangles, and if two triangles are slope triangles then they are similar.
Schaum's Outline of Theory and Problems of Differential and Integral Calculus...Sahat Hutajulu
This document provides the table of contents for the third edition of Schaum's Outline of Theory and Problems of Differential and Integral Calculus. The book contains 76 chapters covering topics in analytic geometry, differential calculus, integral calculus, sequences and series, and multivariable calculus. The third edition has been thoroughly revised with new chapters added on analytic geometry and exponential/logarithmic functions. It aims to provide students with a collection of carefully solved problems representative of those encountered in elementary calculus courses.
The document reports on a lesson about Wilson's theorem and the Chinese remainder theorem. It defines the two theorems, provides examples and proofs of them, and has students work on related activities and problems. It also has students evaluate their understanding of the lesson. Wilson's theorem relates to determining if a number is prime, while the Chinese remainder theorem addresses solving simultaneous congruences.
This document provides an introduction to sets and the real number system. It defines key concepts such as elements, membership, cardinality, types of sets including finite, infinite, empty, subsets, proper subsets, improper subsets, universal sets, power sets, and set relations like equal, equivalent, joint, and disjoint sets. Examples are provided to illustrate each concept. The last section provides a quiz to test understanding of basic set concepts covered.
Cartesian Coordinate Plane - Mathematics 8Carlo Luna
This document explains the Cartesian coordinate plane. It describes how the plane is divided into four quadrants by the x and y axes which intersect at the origin. It provides examples of plotting points using ordered pairs with coordinates (x,y). The document also notes that Rene Descartes developed this system by combining algebra and geometry. It includes an activity for students to physically position themselves on the x and y axes to learn the coordinate system.
This document provides information about eigenvalues and eigenvectors. It defines eigenvalues and eigenvectors as scalars (λ) and vectors (x) that satisfy the equation Ax = λx, where A is a matrix. It discusses properties of eigenvalues including that the sum of eigenvalues is the trace of A, and the product is the determinant. The characteristic equation is defined as det(A - λI) = 0, where the roots are the eigenvalues. Cayley-Hamilton theorem states that every matrix satisfies its own characteristic equation. Examples are given to demonstrate Cayley-Hamilton theorem.
The document summarizes techniques for solving linear systems of equations. It discusses direct solution methods like Gaussian elimination that transform the system into an upper triangular system and then use back substitution to solve. Gaussian elimination involves using elementary row operations to eliminate values below the diagonal of the coefficient matrix. The document also discusses concepts like consistency, uniqueness of solutions, and ill-conditioned systems. It provides examples of applying elementary row operations during the Gaussian elimination process.
Trigonometry deals with relationships between sides and angles in right triangles. The three main trigonometric functions are sine, cosine, and tangent, which provide ratios of sides of a right triangle based on a given angle. A scientific calculator can be used to find trigonometric ratios for any angle, as well as to calculate missing angles or sides of a right triangle when two pieces of information are known.
The document introduces some basic concepts in geometry, including:
1. Points, lines, and planes are undefined terms that form the foundations of geometry.
2. It explains concepts like collinear points, coplanar points, line segments, rays, and how to classify angles.
3. It discusses intersections of lines, planes, and examples of modeling intersections of geometric figures.
Lesson Plan Math : History of measurementxjason001
This document provides an overview of the history of measurement. It discusses how early civilizations used parts of the body to measure things, with the Egyptians using cubits and digits, the Babylonians using minas, and the Chinese using chihs and shihs. Problems arose with this non-standard system, leading to standardized systems like the modern metric system. The document also outlines some key units in other historical systems like those used by the Romans, English, and the eventual development of the International System of Units.
Points, lines, and planes are the basic building blocks of geometry. A point is a location without shape and is represented by a capital letter. A line contains points and has no thickness, with exactly one line passing through any two points. The intersection of two lines is a point. A plane is a flat surface made up of points that extends infinitely in all directions, with the intersection of two planes being a line. Planes are identified by a capital italicized letter or by three non-collinear points.
The document discusses various techniques for instrument drawing and technical lettering. It covers topics such as common drawing equipment like drawing boards, T-squares, and triangles used to draw straight lines and specific angles. It also discusses drawing pencils, lead grades, and their applications for different line types. Specific techniques are covered for drawing horizontal and vertical lines, using a compass to draw circles, and sharpening compass leads. Guidelines are provided for pencil drawing, contrasting line weights, and single-stroke Gothic lettering.
The document discusses sets of axioms and finite geometries. It begins by defining geometry as "earth measure" from its Greek roots and discusses some early examples of geometry like Eratosthenes' measurement of the circumference of the Earth. It then discusses the development of geometry through the Greeks and Euclid, including undefined terms, axioms, postulates, and theorems. It provides examples of Euclid's axioms and postulates and discusses modern refinements. Finally, it introduces finite geometries, which have a finite number of elements, and provides an example of a three-point geometry with its axioms and a proof of one of its theorems.
This video discusses the undefined geometric terms of point, line, and plane. It provides examples of objects that can represent these terms, such as the tip of a needle for a point, an electric wire for a line, and the surface of a table for a plane. The video calls these the "undefined terms" because they do not require further explanation and defines each one as having no dimension for a point, being a set of arranged points for a line, and being a flat endless surface for a plane. It includes classroom activities where students match objects to the terms and examples of how to represent and identify a point, line, and plane.
Math 7 geometry 02 postulates and theorems on points, lines, and planesGilbert Joseph Abueg
This document covers basic concepts in geometry including:
1. Definitions, undefined terms, postulates, and theorems related to points, lines, and planes. Undefined terms include points, lines, and planes. Definitions clearly define concepts like line segments.
2. Postulates are statements accepted as true without proof, including the ruler postulate, segment addition postulate, and plane postulate.
3. Theorems are important statements that can be proven, such as the intersection of lines theorem and the theorem regarding a line and point determining a unique plane.
The document provides information about triangle congruence, including:
1. There are three postulates for proving triangles are congruent: side-side-side (SSS), side-angle-side (SAS), and angle-side-angle (ASA).
2. The SSS postulate states that if three sides of one triangle are congruent to three sides of a second triangle, then the triangles are congruent.
3. The SAS postulate states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the triangles are congruent.
4. The ASA postulate states that if two angles
This document provides information on spherical trigonometry. It defines key terms like great circles, small circles, spherical angles, and spherical triangles. It describes properties of these concepts, such as every great circle passing through the center of a sphere. Formulas for solving spherical triangles are presented, including the Sine Formula, Cosine Formula, and Haversine Formula. Examples show how to use these formulas to calculate unknown sides and angles of spherical triangles given other information.
K TO 12 GRADE 7 LEARNING MODULE IN MATHEMATICS (Quarter 3)LiGhT ArOhL
This document provides a lesson plan on solving linear equations and inequalities in one variable algebraically. It begins with reviewing translating between verbal and mathematical phrases and evaluating expressions. The main focus is on introducing and applying the properties of equality, including reflexive, symmetric, transitive, and substitution properties, to solve equations algebraically. Word problems involving equations in one variable are also discussed. The objectives are to identify and apply the properties of equality to find solutions to equations and solve word problems involving one variable equations.
This document summarizes a Grade 8 mathematics lesson on triangle congruency. The objectives were for students to define and illustrate congruent triangles, identify corresponding sides and angles, and present solutions with accuracy. Content covered triangle congruence. Learning activities included visual aids, measuring triangles with meter sticks, forming pairs of congruent triangles, and applying concepts to a real-world problem of balancing swing supports. Students were evaluated on identifying corresponding parts of congruent triangles. The teacher reflected on students' mastery levels and effectiveness of instructional strategies.
The document provides instructions for organizing and presenting statistical data using frequency tables and histograms. It discusses how to construct a frequency table by grouping raw data into intervals and tallying the frequencies. It then explains how to create a histogram by using the frequency table to draw rectangles whose widths represent intervals and heights represent frequencies. The lesson emphasizes that frequency tables and histograms are useful tools for organizing large data sets and communicating patterns in the data visually.
This document provides information and examples for determining congruence and similarity using slopes of lines and triangles. It explains that the ratio of the rise to the run of two slope triangles formed by a line is equal to the slope of the line. It also gives examples of finding the slope of a roof, stairs, and verifying that the slope is the same between points on the line. Finally, it discusses that right triangles that have their hypotenuses on the same line in a coordinate plane are called slope triangles, and if two triangles are slope triangles then they are similar.
Schaum's Outline of Theory and Problems of Differential and Integral Calculus...Sahat Hutajulu
This document provides the table of contents for the third edition of Schaum's Outline of Theory and Problems of Differential and Integral Calculus. The book contains 76 chapters covering topics in analytic geometry, differential calculus, integral calculus, sequences and series, and multivariable calculus. The third edition has been thoroughly revised with new chapters added on analytic geometry and exponential/logarithmic functions. It aims to provide students with a collection of carefully solved problems representative of those encountered in elementary calculus courses.
The document reports on a lesson about Wilson's theorem and the Chinese remainder theorem. It defines the two theorems, provides examples and proofs of them, and has students work on related activities and problems. It also has students evaluate their understanding of the lesson. Wilson's theorem relates to determining if a number is prime, while the Chinese remainder theorem addresses solving simultaneous congruences.
This document provides an introduction to sets and the real number system. It defines key concepts such as elements, membership, cardinality, types of sets including finite, infinite, empty, subsets, proper subsets, improper subsets, universal sets, power sets, and set relations like equal, equivalent, joint, and disjoint sets. Examples are provided to illustrate each concept. The last section provides a quiz to test understanding of basic set concepts covered.
Cartesian Coordinate Plane - Mathematics 8Carlo Luna
This document explains the Cartesian coordinate plane. It describes how the plane is divided into four quadrants by the x and y axes which intersect at the origin. It provides examples of plotting points using ordered pairs with coordinates (x,y). The document also notes that Rene Descartes developed this system by combining algebra and geometry. It includes an activity for students to physically position themselves on the x and y axes to learn the coordinate system.
This document provides information about eigenvalues and eigenvectors. It defines eigenvalues and eigenvectors as scalars (λ) and vectors (x) that satisfy the equation Ax = λx, where A is a matrix. It discusses properties of eigenvalues including that the sum of eigenvalues is the trace of A, and the product is the determinant. The characteristic equation is defined as det(A - λI) = 0, where the roots are the eigenvalues. Cayley-Hamilton theorem states that every matrix satisfies its own characteristic equation. Examples are given to demonstrate Cayley-Hamilton theorem.
The document summarizes techniques for solving linear systems of equations. It discusses direct solution methods like Gaussian elimination that transform the system into an upper triangular system and then use back substitution to solve. Gaussian elimination involves using elementary row operations to eliminate values below the diagonal of the coefficient matrix. The document also discusses concepts like consistency, uniqueness of solutions, and ill-conditioned systems. It provides examples of applying elementary row operations during the Gaussian elimination process.
This document discusses numerical methods for solving engineering problems. It introduces analytical methods, numerical methods, and experimental methods. It then describes various numerical methods in more detail, including the finite element method (FEM), finite difference method (FDM), finite volume method (FVM), and boundary element method (BEM). It provides examples of the types of problems each method can be applied to and notes the advantages and limitations of each approach.
This document discusses eigen values and eigen vectors of matrices. It defines characteristic matrices, polynomials, and equations. It describes properties of eigen values and lists theorems regarding eigen values and vectors. The Cayley-Hamilton theorem is explained, which states that every square matrix satisfies its own characteristic equation. This theorem can be used to find the inverse and powers of matrices. Diagonalization of matrices is also covered, along with definitions of modal and spectral matrices.
This document discusses several key linear algebra concepts:
1) A square matrix is diagonalizable if it can be transformed into a diagonal matrix through multiplication by an invertible matrix. Diagonalizable matrices can be easily raised to high powers.
2) Eigenvalues and eigenvectors are values and vectors that are unchanged by transformation by the matrix, up to a scaling factor for eigenvectors.
3) Orthogonal matrices preserve lengths and angles when multiplying vectors. The Cayley-Hamilton theorem states that every matrix satisfies its own characteristic equation.
This document discusses solving eigenvalue problems by finding the characteristic equation of matrices and determining their eigenspaces. It provides multiple examples of solving for eigenvalues and eigenspaces of matrices.
B.tech semester i-unit-v_eigen values and eigen vectorsRai University
This document provides information about a course subject and unit. It is for a B.Tech course, the subject is Engineering Mathematics, and the unit is Unit V. The document also specifies that it is from Rai University in Ahmedabad.
A system of linear equations consists of two or more linear equations with two or more variables. The solution to a system is the point where the lines intersect. A system can have 1) one solution, 2) no solution, or 3) infinite solutions. There are three methods to solve a system of linear equations: 1) graphing, 2) elimination, and 3) substitution. The elimination method removes a variable by adding or subtracting the equations while the substitution method replaces a variable in one equation with an expression containing that variable from the other equation.
The document discusses exponential growth functions and provides an example of using an exponential growth equation to model and predict population changes over time. Specifically, it shows how to write an equation modeling the 3.2% annual population growth of Johnson City, TN since 2010, and then uses that equation to calculate that the predicted population in 2020 will be around 34,256 people.
The document presents a system of equations problem about orders placed by a landscaping company. The first order was for 13 bushes and 4 trees totaling $487, and the second order was for 6 bushes and 2 trees totaling $232. The problem is to determine the individual costs of one bush and one tree. Two equations are set up representing the two orders and then solved using the elimination method. The solution is that bushes cost $23 each and trees cost $47 each.
The document discusses singular and non-singular matrices. It defines a singular matrix as a square matrix with a determinant of 0, meaning it is not invertible. A non-singular matrix has a non-zero determinant and is invertible. Examples of singular matrices include matrices with a row or rows of all zeros, equal rows, or an eigenvector of 0. Non-singular matrices have determinants not equal to 0 and include strictly diagonal dominant matrices. The comparison section outlines key differences between singular and non-singular matrices.
This document provides an overview of a quantum mechanics course taught by Martin Plenio at Imperial College in 2002. The course covers mathematical foundations of quantum mechanics, quantum measurements, dynamics and symmetries, and approximation methods. It is divided into two parts, with the first part covering core topics in quantum mechanics and the second part focusing on quantum information processing and related topics. The document provides chapter outlines and section headings for the material to be covered.
This document provides biographical details about several prominent 20th century physicists:
- Emilio Segre studied x-rays and helped develop modern particle physics. He was invited to the 5th Solvay Conference due to his discoveries.
- Louis de Broglie proposed that particles like electrons exhibit both wave-like and particle-like properties. This helped lay the foundations for quantum mechanics.
- Werner Heisenberg, Max Born, and Pascual Jordan developed matrix mechanics, an early formulation of quantum mechanics using non-commutative matrices.
- Wolfgang Pauli made important contributions including the exclusion principle and hypothesis of neutrino and nuclear spin.
This document discusses methods for solving systems of linear equations. It covers direct methods like Gaussian elimination and LU factorization. Gaussian elimination reduces a system of equations to upper triangular form using elementary row operations. LU factorization expresses the coefficient matrix as the product of a lower triangular matrix and an upper triangular matrix. The document provides examples to demonstrate Gaussian elimination with partial pivoting and solving a system using LU factorization. Iterative methods are also introduced as an alternative to direct methods for large systems.
Applications of the surface finite element methodtr1987
A coupled bulk-surface finite element method is presented to solve problems arising in cell biology. Optimal order estimates for a linear elliptic equation are shown along with some numerical examples. An example of a parabolic problem with nonlinear coupling governed by Langmuir kinetics is presented, which describes the process of fluorescence recovery after photo bleaching (FRAP) in biological cells.
This document discusses mathematical methods for matrices and linear algebra. It covers topics like matrices and linear systems of equations, eigenvalues and eigenvectors, real and complex matrices and quadratic forms. It also discusses algebraic and transcendental equations, interpolation, curve fitting, numerical differentiation and integration, numerical solutions to ordinary differential equations, Fourier series and transforms, and partial differential equations. It provides references to several textbooks and other resources on these mathematical methods.
The document discusses face recognition using eigen values extracted from principal component analysis of multiple face images and feeding those values into a neural network. It describes extracting eigen values from a dataset of 8 persons with 10 images each. The neural network has an input layer of 10 neurons, 3 hidden layers of 10 neurons each, and an output layer of 1 neuron. Eigenfaces are extracted from the training set to define the face space. The neural network is trained on the eigen values and tested to match faces by finding the least error.
This document summarizes key concepts regarding eigenvalues and eigenvectors of matrices:
- Eigenvalues are scalars such that there exist non-zero eigenvectors satisfying Ax = λx.
- The characteristic equation states that λ is an eigenvalue if and only if it satisfies det(A - λI) = 0.
- A matrix is diagonalizable if it can be written as A = PDP-1, where D is a diagonal matrix of eigenvalues and P is a matrix of corresponding eigenvectors. Diagonalizable matrices can easily compute powers by raising the eigenvalues to powers.
Difference between Classical Physics and Quantum Mechanics. I have presented different types of double-slit experiments to proof superposition. Then, there is an explanation of Shrodinger's Cat theoretical experiment using animation.
This document discusses the rank of matrices and how it relates to the solvability of linear systems of equations. It contains the following key points:
1) The rank of a matrix is the number of leading entries in its row-reduced form and determines the number of independent variables in a linear system with that matrix as its coefficient matrix.
2) The rank of the coefficient matrix and augmented matrix determine whether a linear system has no solution, a unique solution, or infinitely many solutions.
3) Homogeneous systems always have at least one solution (the trivial solution of all zeros) and the rank of the coefficient matrix determines if that is the only solution or if there are infinitely many solutions.
The document provides an introduction to solving systems of linear equations using Gaussian elimination. It defines linear systems and the matrix formulation. Gaussian elimination transforms the coefficient matrix into upper triangular form through elementary row operations. The method is then demonstrated on an example system. Key steps include choosing pivots, making entries below the pivot zero, and ultimately solving the system using back substitution.
1. Linear Algebra for Machine Learning: Linear SystemsCeni Babaoglu, PhD
The seminar series will focus on the mathematical background needed for machine learning. The first set of the seminars will be on "Linear Algebra for Machine Learning". Here are the slides of the first part which is giving a short overview of matrices and discussing linear systems.
This document provides an introduction to systems of linear equations and matrix operations. It defines key concepts such as matrices, matrix addition and multiplication, and transitions between different bases. It presents an example of multiplying two matrices using NumPy. The document outlines how systems of linear equations can be represented using matrices and discusses solving systems using techniques like Gauss-Jordan elimination and elementary row operations. It also introduces the concepts of homogeneous and inhomogeneous systems.
Linear Algebra may be defined as the form of algebra in which there is a study of different kinds of solutions which are related to linear equations. In order to explain the Linear Algebra, it is important to explain that the title consists of two different terms. The very first term which is important to be considered in the same, is Linear. Linear may be defined as something which is straight. Linear equations can be used for the calculation of the equation in a xy plane where the straight lines has been defined. In addition to this, linear equations can be used to define something which is straight in a three dimensional perspective. Another view of linear equations may be defined as flatness which recognizes the set of points which can be used for giving the description related to the equations which are in a very simple forms. These are the equations which involves the addition and multiplication.
A system of linear equations in two variables can be solved either graphically or algebraically. Graphically involves drawing the lines and finding their point of intersection. Algebraically involves combining the equations to eliminate variables until one is left. A system has a single solution if the lines intersect, no solution if parallel, and infinite solutions if coincident. Algebraic methods include substitution and elimination to solve for the variables.
Chapter 3: Linear Systems and Matrices - Part 1/SlidesChaimae Baroudi
The document provides information about linear systems and matrices. It begins by defining linear and non-linear equations. It then discusses systems of linear equations, their graphical and geometric interpretations, and the three possible solutions: no solution, a unique solution, or infinitely many solutions. The document also covers matrix notation for representing linear systems, elementary row operations for transforming systems, and determining whether a system has a solution and whether that solution is unique.
A system of linear equations in two variables can be solved either graphically or algebraically. Graphically, the solutions are found by drawing the lines corresponding to each equation and finding their point(s) of intersection. Algebraically, the equations are combined to eliminate one variable, resulting in an equation that can be solved for the remaining variable. A system has a single solution if the lines intersect at one point, no solution if the lines are parallel, or infinite solutions if the lines coincide as the same line.
A system of linear equations in two variables can be solved either graphically or algebraically. Graphically, the solutions are found by drawing the lines corresponding to each equation and finding their point(s) of intersection. Algebraically, the equations are combined to eliminate one variable, resulting in an equation that can be solved for the remaining variable. A system has a single solution if the lines intersect at one point, no solution if the lines are parallel, or infinite solutions if the lines coincide as the same line.
1) The graphical method involves graphing the lines represented by each equation on the same coordinate plane and finding the point where they intersect, which gives the solution.
2) Cramer's rule expresses each unknown as a ratio of determinants, with the numerator being the determinant of the coefficients with one column replaced by constants.
3) Gaussian elimination transforms the matrix of coefficients into upper triangular form using elementary row operations, then back substitution can be used to solve for the unknowns.
1) The graphical method involves graphing the lines represented by each equation on the same coordinate plane and finding the point where they intersect, which gives the solution.
2) Cramer's rule expresses each unknown as a ratio of determinants, with the numerator being the determinant of the coefficient matrix with one column replaced by the constants.
3) Gaussian elimination transforms the coefficient matrix into upper triangular form using elementary row operations, then back substitution solves for the unknowns.
This document discusses various direct methods for solving linear systems of equations, including graphical methods, Cramer's rule, elimination of unknowns, Gaussian elimination, Gaussian-Jordan elimination, and LU decomposition. It provides examples and explanations of each method. Graphical methods can solve systems of 2 equations visually by plotting the lines. Cramer's rule uses determinants to find solutions. Elimination of unknowns combines equations to remove variables. Gaussian elimination converts the matrix to upper triangular form. Gaussian-Jordan elimination converts it to an identity matrix. LU decomposition factors the matrix into lower and upper triangular matrices.
1) The graphical method involves graphing the lines represented by each equation on the same coordinate plane and finding the point where they intersect, which gives the solution.
2) Cramer's rule expresses each unknown as a ratio of determinants, with the numerator being the determinant of the coefficient matrix with one column replaced by the constants.
3) Gaussian elimination transforms the coefficient matrix into upper triangular form using elementary row operations, then back substitution solves for the unknowns.
This document provides an overview and definitions of key concepts from Chapter 1 of a college mathematics textbook, including: linear equations and inequalities in standard form and how they are solved; the Cartesian coordinate system and how graphs of linear equations form lines; determining the slope and equations of lines in slope-intercept and point-slope form; the relationship between supply and demand curves; and using linear regression to fit a line to scatter plot data and make predictions.
Linear Algebra Presentation including basic of linear AlgebraMUHAMMADUSMAN93058
This document discusses linear algebra concepts including systems of linear equations, matrices, and matrix operations. It covers topics such as matrix addition, subtraction, multiplication, and transposition. Matrix-vector products and partitioned matrices are also explained. Elementary row operations are defined as interchanging rows, multiplying a row by a non-zero number, and adding a multiple of one row to another. The document concludes by defining row reduced echelon form (RREF) and row echelon form (REF) of a matrix.
1. Homework Task 3 on systems of equations and inequalities is due on August 6. Students should check all memos on the online learning platform.
2. The document discusses finding inverses of matrices and solving matrix equations. It provides examples of finding the inverse of 2x2 and 3x3 matrices using elementary row operations to transform the matrices into an identity matrix.
3. Solving a system of equations using a matrix inverse involves writing the system as a matrix equation AX=B, then multiplying both sides by the inverse of the coefficient matrix A to isolate the solution vector X.
The document discusses systems of linear equations and Gaussian elimination. It begins with an introduction to systems of linear equations, defining them as a set of linear equations that can have no solution, a unique solution, or infinitely many solutions. It then discusses row-echelon form and reduced row-echelon form, which are special forms that a matrix of a linear system can be put into using elementary row operations. Being in row-echelon or reduced row-echelon form provides information about the solution set of the corresponding linear system. Examples are provided to illustrate these concepts.
Matrices and System of Linear Equations pptDrazzer_Dhruv
The document discusses matrices and systems of linear equations. It defines matrices and different types of matrices including square, diagonal, scalar, identity, zero, negative, upper triangular, lower triangular, and transpose matrices. It also covers properties of matrix operations and examples of finding the transpose of matrices. The document then discusses row echelon form (REF) and reduced row echelon form (RREF) as well as the different types of solutions that systems of linear equations can have.
The document presents information on matrices, including:
- Definitions of matrices as rectangular arrangements of numbers arranged in rows and columns
- Common matrix operations such as addition, subtraction, scalar multiplication, and matrix multiplication
- Determinants and inverses of matrices
- How matrices can represent systems of linear equations
- Unique properties of matrices, such as the product of two non-zero matrices possibly being zero
- Applications of matrices in fields like geology, statistics, economics, and animation
The document provides an overview of matrix theory, including:
1. The definition and notation of matrices, including that a matrix A is represented as Am×n, where m is the number of rows and n is the number of columns.
2. The different types of matrices and operations that can be performed on matrices, such as scalar multiplication, matrix multiplication, and properties like the distributive law.
3. Methods for solving systems of linear equations using matrices, including writing the system in matrix form, reducing the augmented matrix to echelon form, and determining the solution based on the rank.
1. Solution of System of Linear Equations
&
Eigen Values and Eigen Vectors
P. Sam Johnson
March 30, 2015
P. Sam Johnson (NITK) Solution of System of Linear Equations & Eigen Values and Eigen Vectors March 30, 2015 1/43
2. Overview
Linear systems
Ax = b (1)
occur widely in applied mathematics.
They occur as direct formulations of “real world” problems ; but more
often, they occur as a part of numerical analysis.
There are two general categories of numerical methods for solving (1).
Direct methods are methods with a finite number of steps ; and they end
with the exact solution provided that all arithmetic operations are exact.
Iterative methods are used in solving all types of linear systems, but they
are most commonly used for large systems.
P. Sam Johnson (NITK) Solution of System of Linear Equations & Eigen Values and Eigen Vectors March 30, 2015 2/43
3. Overview
Iterative methods are faster than the direct methods. Even the round-off
errors in iterative methods are smaller.
In fact, iterative method is a self correcting process and any error made
at any stage of computation gets automatically corrected in the
subsequent steps.
We start with a simple example of a linear system and we discuss
consistency of system of linear equations, using the concept of
determinant.
Two iterative methods – Gauss-Jacobi and Gauss-Seidel methods are
discussed, to solve a given linear system numerically.
P. Sam Johnson (NITK) Solution of System of Linear Equations & Eigen Values and Eigen Vectors March 30, 2015 3/43
4. Overview
Eigen values are a special set of scalars associated with a linear system of
equations. The determination of the eigen values and eigen vectors of a
system is extremely important in physics and engineering, where it is
equivalent to matrix diagonalization and arises in such common
applications as stability analysis, the physics of rotating bodies, and small
oscillations of vibrating systems, to name only a few.
Each eigen value is paired with a corresponding vector, called eigen
vector.
We discuss properties of eigen values and eigen vectors in the lecture. Also
we give a method, called Rayeigh’s Power Method, to find out the
largest eigen value in magnitude (also called, dominant eigen value). The
special advantage of the power method is that the eigen vector corresponds
to the dominant eigen value is also calculated at the same time.
P. Sam Johnson (NITK) Solution of System of Linear Equations & Eigen Values and Eigen Vectors March 30, 2015 4/43
5. An Example
A shopkeeper offers two standard packets because he is convinced that
north indians each more wheat than rice and south indians each more rice
than wheat. Packet one P1 : 5kg wheat and 2kg rice ; Packet two P2 :
2kg wheat and 5kg rice. Notation. (m, n) : m kg wheat and n kg rice.
Suppose I need 19kg of wheat and 16kg of rice. Then I need to buy x
packets of P1 and y packets of P2 so that x(5, 2) + y(2, 5) = (10, 16).
Hence I have to solve the following system of linear equations
5x + 2y = 10
2x + 5y = 16.
Suppose I need 34 kg of wheat and 1 kg of rice. Then I must buy 8
packets of P1 and −3 packets of P2. What does this mean? I buy 8
packets of P1 and from these I make three packets of P2 and give them
back to the shopkeeper.
P. Sam Johnson (NITK) Solution of System of Linear Equations & Eigen Values and Eigen Vectors March 30, 2015 5/43
6. Consistency of System of Linear Equations – Graphically
(1 Equation, 2 Variables)
The solution of the equation
ax + by = c
is the set of all points satisfy the equation forms a straight line in the
plane through the point (c/b, 0) and with slope −a/b.
Two lines
parallel – no solution
intersect – unique solution
same – infinitely many solutions.
P. Sam Johnson (NITK) Solution of System of Linear Equations & Eigen Values and Eigen Vectors March 30, 2015 6/43
7. Consistency of System of Linear Equations
Consider the system of m linear equations in n unkowns
a11x1 + a12x2 + · · · + a1nxn = b1
a21x1 + a22x2 + · · · + a2nxn = b2
... +
... + · · · +
... =
...
am1x1 + am2x2 + · · · + amnxn = bm.
Its matrix form is
a11 a12 . . . a1n
a21 a22 . . . a2n
...
... . . .
...
am1 am2 . . . amn
x1
x2
...
xm
=
b1
b2
...
bm.
That is, Ax = b, where A is called the coefficient matrix.
P. Sam Johnson (NITK) Solution of System of Linear Equations & Eigen Values and Eigen Vectors March 30, 2015 7/43
8. To determine whether these equations are consistent or not, we find the
ranks of the coefficient matrix A and the augmented matrix
K =
a11 a12 . . . a1n b1
a21 a22 . . . a2n b2
...
... · · ·
...
...
am1 am2 . . . amn bm
= [A b].
We denote the rank of A by r(A).
1. r(A) = r(K), then the linear system Ax = b is inconsistent and has
no solution.
2. r(A) = r(K) = n, then the linear system Ax = b is consistent and
has a unique solution.
3. r(A) = r(K) < n, then the linear system Ax = b is consistent and
has an infinite number of solutions.
P. Sam Johnson (NITK) Solution of System of Linear Equations & Eigen Values and Eigen Vectors March 30, 2015 8/43
9. Solution of Linear Simultaneous Equations
Simultaneous linear equations occur quite often in engineering and science.
The analysis of electronic circuits consisting of invariant elements, analysis
of a network under sinusoidal steady-state conditions, determination of the
output of a chemical plant, finding the cost of chemical reactions are some
of the problems which depend on the solution of simultaneous linear
algebraic equations, the solution of such equations can be obtained by
direct or iterative methods (successive approximation methods).
Some direct methods are as follows:
1. Method of determinants – Cramer’s rule
2. Matrix inversion method
3. Gauss elimination method
4. Gauss-Jordan method
5. Factorization (triangulization) method.
P. Sam Johnson (NITK) Solution of System of Linear Equations & Eigen Values and Eigen Vectors March 30, 2015 9/43
10. Iterative Methods
Direct methods yield the solution after a certain amount of computation.
On the other hand, an iterative method is that in which we start from an
approximation to the true solution and obtain better and better
approximations from a computation cycle repeated as often as may be
necessary for achieving a desired accuracy.
Thus in an iterative method, the amount of computation depends on the
degree of accuracy required.
For large systems, iterative methods are faster than the direct methods.
Even the round-off errors in iterative methods are smaller. In fact,
iteration is a self correcting process and any error made at any stage of
computation gets automatically corrected in the subsequent steps.
P. Sam Johnson (NITK) Solution of System of Linear Equations & Eigen Values and Eigen Vectors March 30, 2015 10/43
11. Diagonally Dominant
Definition
An n × n matrix A is said to be diagonally dominant if, in each row, the
absolute value of each leading diagonal element is greater than or equal to
the sum of the absolute values of the remaining elements in that row.
The matrix A =
10 −5 −2
4 −10 3
1 6 10
is a diagonally dominant and the
matrix A =
2 3 −1
5 8 −4
1 1 1
is not a diagonally dominant matrix.
P. Sam Johnson (NITK) Solution of System of Linear Equations & Eigen Values and Eigen Vectors March 30, 2015 11/43
12. Diagonal System
Definition
In the sytem of simultaneous linear equations in n unknowns AX = B, if A
is diagonally dominant, then the system is said to be diagonal system.
Thus the system of linear equations
a1x + b1y + c1z = d1
a2x + b2y + c2z = d2
a3x + b3y + c3z = d3
is a diagonal system if
|a1| ≥ |b1| + |c1|
|b2| ≥ |a2| + |c2|
|c3| ≥ |a3| + |b3|.
P. Sam Johnson (NITK) Solution of System of Linear Equations & Eigen Values and Eigen Vectors March 30, 2015 12/43
13. The process of iteration in solving AX = B will converge quickly if the
cofficient matrix A is diagonally dominant.
If the coefficient matrix is not diagonally dominant we must rearrange the
equations in such a way that the resulting coefficient matrix becomes
dominant, if possible, before we apply the iteration method.
Exercises
1. Is the system of equations diagonally dominant? If not, make it diagonally dominant.
3x + 9y − 2z = 10
4x + 2y + 13z = 19
4x − 2y + z = 3.
2. Check whether the system of equations
x + 6y − 2z = 5
4x + y + z = 6
−3x + y + 7z = 5.
is a diagonal system. If not, make it a diagonal system.
P. Sam Johnson (NITK) Solution of System of Linear Equations & Eigen Values and Eigen Vectors March 30, 2015 13/43
14. Jacobi Iteration Method (also known as Gauss-Jacobi)
Simple iterative methods can be designed for systems in which the
coefficients of the leading diagonal are large as comparted to others.
We now describe three such methods. Gauss-Jacobi Iteration Method
Consider the system of linear equations
a1x + b1y + c1z = d1
a2x + b2y + c2z = d2
a3x + b3y + c3z = d3.
If a1, b2, c3 are large as compared to other coefficients (if not, rearrange
the given linear equations), solve for x, y, z respectively. That is, the
cofficient matrix is diagonally dominant.
P. Sam Johnson (NITK) Solution of System of Linear Equations & Eigen Values and Eigen Vectors March 30, 2015 14/43
15. Then the system can be written as
x =
1
a1
(d1 − b1y − c1z)
y =
1
b2
(d2 − a2x − c2z) (2)
z =
1
c3
(d3 − a3x − b3y).
Let us start with the initital approximations x0, y0, z0 for the values of
x, y, z repectively. In the absence of any better estimates for x0, y0, z0,
these may each be taken as zero.
Substituting these on the right sides of (2), we gt the first approximations.
This process is repeated till the difference between two consecutive
approximations is negligible.
P. Sam Johnson (NITK) Solution of System of Linear Equations & Eigen Values and Eigen Vectors March 30, 2015 15/43
16. Exercises
3. Solve, by Jacobi’s iterative method, the equations
20x + y − 2z = 17
3x + 20y − z = −18
2x − 3y + 20z = 25.
4. Solve, by Jacobi’s iterative method correct to 2 decimal places, the equations
10x + y − z = 11.19
x + 10y + z = 28.08
−x + y + 10z = 35.61.
5. Solve the equations, by Gauss-Jacobi iterative method
10x1 − 2x2 − x3 − x4 = 3
−2x1 + 10x2 − x3 − x4 = 15
−x1 − x2 + 10x3 − 2x4 = 27
−x1 − x2 − 2x3 + 10x4 = −9.
P. Sam Johnson (NITK) Solution of System of Linear Equations & Eigen Values and Eigen Vectors March 30, 2015 16/43
17. Gauss-Seidel Iteration Method
This is a modification of Jacobi’s Iterative Method.
Consider the system of linear equations
a1x + b1y + c1z = d1
a2x + b2y + c2z = d2
a3x + b3y + c3z = d3.
If a1, b2, c3 are large as compared to other coefficients (if not, rearrange
the given linear equations), solve for x, y, z respectively.
Then the system can be written as
x =
1
a1
(d1 − b1y − c1z)
y =
1
b2
(d2 − a2x − c2z) (3)
z =
1
c3
(d3 − a3x − b3y).
P. Sam Johnson (NITK) Solution of System of Linear Equations & Eigen Values and Eigen Vectors March 30, 2015 17/43
18. Here also we start with the initial approximations x0, y0, z0 for x, y, z
respectively, (each may be taken as zero).
Subsituting y = y0, z = z0 in the first of the equations (3), we get
x1 =
1
a1
(d1 − b1y0 − c1z0).
Then putting x = x1, z = z0 in the second of the equations (3), we get
y1 =
1
b2
(d2 − a2x1 − c2z0).
Next subsituting x = x1, y = y1 in the third of the equations (3), we get
z1 =
1
c3
(d3 − a3x1 − b3y1)
and so on.
P. Sam Johnson (NITK) Solution of System of Linear Equations & Eigen Values and Eigen Vectors March 30, 2015 18/43
19. That is, as soon as a new approximation for an unknown is found, it is
immediately used in the next step.
This process of iteration is repeatedly till the values of x, y, z are obtained
to desired degree of accuracy.
Since the most recent approximations of the unknowns are used which
proceeding to the next step, the convergence in the Gauss-Seidel
method is twice as fast as in Gauss-Jacobi method.
P. Sam Johnson (NITK) Solution of System of Linear Equations & Eigen Values and Eigen Vectors March 30, 2015 19/43
20. Convergence Criteria
The condition for convergence of Gauss-Jacobi and Gauss-Seidel methods
is given by the following rule:
The process of iteration will converge if in each equation of the system,
the absolute value of the largest co-efficient is greater than the sum of the
absolute values of all the remaining coefficients.
That is, if the coefficient matrix is a diagonal dominant matrix, then the
process of iteration will converge.
For a given system of linear equations, before finding approximations by
using Gauss-Jacobi / Gauss-Seidel methods, convergence criteria should be
verified. If the coefficient matrix is a not diagonal dominant matrix,
rearrange the equations, so that the new coefficient matrix is diagonal
dominant. Note that all systems of linear equations are not diagonal
dominant.
P. Sam Johnson (NITK) Solution of System of Linear Equations & Eigen Values and Eigen Vectors March 30, 2015 20/43
21. Exercises
6. Apply Gauss-Seidel iterative method to solve the equations
20x + y − 2z = 17
3x + 20y − z = −18
2x − 3y + 20z = 25.
7. Solve the equations, by Gauss-Jacobi and Gauss-Seidel methods (and compare the values)
27x + 6y − z = 85
x + y + 54z = 110
6x + 15y + 2z = 72.
8. Apply Gauss-Seidel iterative method to solve the equations
10x1 − 2x2 − x3 − x4 = 3
−2x1 + 10x2 − x3 − x4 = 15
−x1 − x2 + 10x3 − 2x4 = 27
−x1 − x2 − 2x3 + 10x4 = −9.
P. Sam Johnson (NITK) Solution of System of Linear Equations & Eigen Values and Eigen Vectors March 30, 2015 21/43
22. Relaxation Method
Consider the system of linear equations
a1x + b1y + c1z = d1
a2x + b2y + c2z = d2
a3x + b3y + c3z = d3.
We define the residuals Rx , Ry , Rz by the relations
Rx = d1 − a1x − b1y − c1z
Ry = d2 − a2x − b2y − c2z (4)
Rz = d3 − a3x − b3y − c3z.
P. Sam Johnson (NITK) Solution of System of Linear Equations & Eigen Values and Eigen Vectors March 30, 2015 22/43
23. To start with we assume that x = y = z = 0 and calculate the initial
residuals.
Then the residuals are reduced step by step, by giving increments to
the variables.
We note from the equations (4) that if x is increased by 1 (keeping y and
z constant), Rx , Ry and Rz decrease by a1, a2, a3 respectively. This is
shown in the following table along with the effects on the residuals when y
and z are given unit increments.
δRx δRy δRz
δx = 1 −a1 −a2 −a3
δy = 1 −b1 −b2 −b3
δz = 1 −c1 −c2 −c3
The table is the negative of transpose of the coefficient matrix.
P. Sam Johnson (NITK) Solution of System of Linear Equations & Eigen Values and Eigen Vectors March 30, 2015 23/43
24. At each step, the numerically largest residual is reduced to almost zero.
How can one reduce a particular residual?
To reduce a particular residual, the value of the corresponding variable is
changed.
For example, to reduce Rx by p, x should be increased by p/a1.
When all the residuals have been reduced to almost zero, the increments
in x, y, z are added separately to give the desired solution.
Verification Process : The residuals are not all negligible when the
computed values of x, y, z are substituted in (4), then there is some
mistake and the entire process should be rechecked.
P. Sam Johnson (NITK) Solution of System of Linear Equations & Eigen Values and Eigen Vectors March 30, 2015 24/43
25. Convergence of the Relaxation Method
Relaxation method can be applied successfully only if there is at least one
row in which diagonal element of the coefficient matrix dominate the
other coefficients in the corresponding row.
That is,
|a1| ≥ |b1| + |c1|
|b2| ≥ |a2| + |c2|
|c3| ≥ |a3| + |b3|
should be valid for at least one row.
P. Sam Johnson (NITK) Solution of System of Linear Equations & Eigen Values and Eigen Vectors March 30, 2015 25/43
26. Exercises
9. Solve by relaxation method, the equations
9x − 2y + z = 50
x + 5y − 3z = 18
−2x + 2y + 7z = 19.
10. Solve by relaxation method, the equations
10x − 2y − 3z = 205
−2x + 10y − 2z = 154
−2x − y + 10z = 120.
P. Sam Johnson (NITK) Solution of System of Linear Equations & Eigen Values and Eigen Vectors March 30, 2015 26/43
27. Eigen Values and Eigen Vectors
For any n × n matrix A, consider the polynomial
χA(λ) := |λI − A| =
λ − a11 −a12 · · · −a1n
−a21 λ − a22 · · · −a2n
· · · · · · · · · · · ·
−an1 −an2 · · · λ − ann
. (5)
Clearly this is a monic polynomial of degree n. By the fundamental
theorem of algebra, χ(A) has exactly n (not necessarily distinct) roots.
χA(λ) the characteristic polynomial of
A
χA(λ) = 0 the characteristic equation of A
the roots of χA(λ) the characteristic roots of A
distinct roots of χA(λ) the spectrum of A
P. Sam Johnson (NITK) Solution of System of Linear Equations & Eigen Values and Eigen Vectors March 30, 2015 27/43
28. Few Observations on Characteristic Polynomials
The constant terms and the coefficient of λn−1 in χA(λ) are (−1)n|A|
and tr(A).
The sum of the characteristic roots of A is tr(A) and the product of
the characteristic roots of A is |A|.
Since λI − AT = (λI − A)T , characteristic polynomials of A and AT
are the same.
Since λI − P−1AP = P−1(λI − A)P, similar matrices have the same
characteristic polynomials.
P. Sam Johnson (NITK) Solution of System of Linear Equations & Eigen Values and Eigen Vectors March 30, 2015 28/43
29. Eigen Values and Eigen Vectors
Definition
A complex number λ is an eigen value of A if there exists x = 0 in Cn
such that Ax = λx. Any such (non-zero) x is an eigen vector of A
corresponding to the eigen value λ.
When we say that x is an eigen vector of A we mean that x is an eigen
vector of A corresponding to some eigen value of A.
λ is an eigen value of A iff the system (λI − A)x = 0 has a non-trivial
solution.
λ is a characteristic root of A iff λI − A is singular.
Theorem
A number λ is an eigen value of A iff λ is a characteristic root of A.
P. Sam Johnson (NITK) Solution of System of Linear Equations & Eigen Values and Eigen Vectors March 30, 2015 29/43
30. The preceding theorem shows that eigen values are the same as
characteristic roots. However, by ‘the characteristic roots of A’ we mean
the n roots of the characteristic polynomial of A whereas ‘the eigen values
of A’ would mean the distinct characteristic roots of A.
Equivalent names:
Eigen values proper values, latent roots, etc.
Eigen vectors characteristic vectors, latent vectors, etc.
P. Sam Johnson (NITK) Solution of System of Linear Equations & Eigen Values and Eigen Vectors March 30, 2015 30/43
31. Properties of Eigen Values
1. The sum and product of all eigen values of a matrix A is the sum of
principal diagonals and determinant of A, respectively. Hence a
matrix is not invertible if and only if it has a zero eigen value.
2. The eigen values of A and AT (the transpose of A) are same.
3. If λ1, λ2, . . . , λn are eigen values of A, then
(a) 1/λ1, 1/λ2, . . . , 1/λn are eigen values of A−1
.
(b) kλ1, kλ2, . . . , kλn are eigen values of kA.
(c) kλ1 + m, kλ2 + m, . . . , kλn + m are eigen values of kA + mI.
(d) λp
1, λp
2, . . . , λp
n are eigen values of Ap
, for any positive integer p.
4. The transformation of A by a non-singular matrix P to P−1AP is
called a similarity transformation.
Any similarity transformation applied to a matrix leaves its eigen
values unchanged.
5. The eigen values of a real symmetric matrix are real.
P. Sam Johnson (NITK) Solution of System of Linear Equations & Eigen Values and Eigen Vectors March 30, 2015 31/43
32. Cayley - Hamilton Theorem
Theorem
For every matrix A, the characteristic polynomial of A annihilates A.
That is, every matrix satisfies its own characteristic equation.
Main uses of Cayley-Hamilton theorem
1. To evaluate large powers A.
2. To evaluate a polynomial in A with large degree even if A is singular.
3. To express A−1 as a polynomial in A whereas A is non-singular.
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33. Dominant Eigen Value
We have seen that the eigen values of an n × n matrix A are obtained by
solving its characteristic equation
λn
+ an−1λn−1
+ cn−2λn−2
+ · · · + c0 = 0.
For large values of n, polynomial equations like this one are difficult and
time-consuming to solve.
Moreover, numerical techniques for approximating roots of polynomial
equations of high degree are sensitive to rounding errors. We now look at
an alternative method for approximating eigen values.
As presented here, the method can be used only to find the eigen
value of A, that is, largest in absolute value – we call this eigen value the
dominant eigen value of A.
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34. Dominant Eigen Vector
Dominant eigen values are of primary interest in many physical
applications.
Let λ1, λ2, . . . , and λn be the eigen values of an n × n matrix A.
λ1 is called the dominant eigen value of A if
|λi | > |λj |, i = 2, 3, . . . , n.
For example, the matrix
1 0
0 −1
has no dominant eigen value.
The eigen vectors corresponding to λ1 are called dominant eigen vectors
of A.
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35. Power Method
In many engineering problems, it is required to compute the numerically
largest eigen values (dominant eigen values) and the corresponding eigen
vectors (dominant eigen vectors).
In such cases, the following iterative method is quite convenient which is
also well-suited for machine computations.
Let x1, x2, . . . , xn be the eigen vectors corresponding to the eigen values
λ1, λ2, . . . , λn.
We assume that the eigen values of A are real and distint with
|λ1| > |λ2| > · · · > |λn|.
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36. As the vectors x1, x2, . . . , xn are linearly independent, any arbitrary column
vector can be written as
x = k1x1 + k2x2 + · · · + knxn.
We now operate A repeatedly on the vector x.
Then
Ax = k1λ1x1 + k2λ2x2 + · · · + knλnxn.
Hence
Ax = λ1 k1x1 + k2
λ2
λ1
x2 + · · · + kn
λn
λ1
xn .
and through iteration we obtain,
Ar
x = λ1 k1x1 + k2
λ2
λ1
r
x2 + · · · + kn
λn
λ1
r
xn (6)
provided λ1 = 0.
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37. Since λ1 is dominant, all terms inside the brackets have limit zero except
the first term.
That is, for large values of r, the vector
k1x1 + k2
λ2
λ1
r
x2 + · · · + kn
λn
λ1
r
xn
will converge to k1x1, that is the eigen vector of λ1.
The eigenvalue is obtained as
λ1 = lim
n→∞
(Ar+1x)p
(Ar x)p
p = 1, 2, . . . , n
where the index p signifies the pth component in the corresponding vector.
That is, if we take the ratio of any corresponding components of Ar+1x
and Ar x, this ratio should therefore have a limit λ1.
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38. Linear Convergence
Neglecting the terms of λ2, λ3, . . . , λn in (6) we get
Ar+1x
k1λr+1
1
− x1 =
λ2
λ1
Ar x
k1λr
1
− x1
which shows that the convergence is linear with convergence ratio
λ2
λ1
.
Thus, the convergence is faster than if this ratio is small.
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39. Working Rule To Find Dominant Eigen Value
Choose x = 0, an arbitrary real vector.
Generally, we choose x as
1
1
1
, or,
1
0
0
.
In other words, for the purpose of computation, we generally take x with 1
as the first component.
We start with a column vector x which is as near the solution as possible
and evaluate Ax which is written as
λ(1)
x(1)
after normalization. This gives the first approximation λ(1) to the eigen
value and x(1) is the eigen vector.
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40. Rayeigh’s Power Method
Similarly, we evaluate Ax(2) = λ(2)x(2) which gives the second
approximation. We repeat this process till [x(r) − x(r−1)] becomes
negligible.
Then λ(r) will be the dominant (largest) eigen value and x(r), the
corresponding eigen vector (dominant eigen vector).
The iteration method of finding the dominant eigen value of a square
matrix is called the Rayeigh’s power method.
At each stage of iteration, we take out the largest component from y(r) to
find x(r) where
y(r+1)
= Ax(r)
, r = 0, 1, 2, . . . .
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41. To find the Smallest Eigen Value
To find the smallest eigen value of a given matrix A, apply power method
to the matrix A−1.
Find the dominant eigen value of A−1 and then take is reciprocal.
Exercise
11. Say true or false with justification: If λ is the dominant eigen value of
A and β is the dominant eigen value of A − λI, then the smallest
eigen value of A is λ + β.
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42. Exercises
12. Determine the largest eigen value and the corresponding eigen vector
of the matrix
5 4
1 2
.
13. Find the largest eigen value and the corresponding eigen vector of the
matrix
2 −1 0
−1 2 −1
0 −1 2
using power method. Take [1, 0, 0]T as an initial eigen vector.
14. Obtain by power method, the numerically dominant eigen value and
eigen vector of the matrix
15 −4 −3
−10 12 −6
−20 4 −2.
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43. References
1. Richard L. Burden and J. Douglas Faires, Numerical Analysis -
Theory and Applications, Cengage Learning, Singapore.
2. Kendall E. Atkinson, An Introduction to Numerical Analysis, Wiley
India.
3. David Kincaid and Ward Cheney, Numerical Analysis -
Mathematics of Scientific Computing, American Mathematical
Society, Providence, Rhode Island.
4. S.S. Sastry, Introductory Methods of Numerical Analysis, Fourth
Edition, Prentice-Hall, India.
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