This document provides an overview of the content that will be covered in a session on teaching matrices for the Edexcel FP3 module. The session will begin with an introduction to matrices, including examples and definitions. It will then cover linear transformations represented by matrices, showing how matrices can represent rotations. Subsequent sections will address the transpose and inverse of matrices, eigenvalues and eigenvectors, and symmetric matrices. The session will conclude with examples of typical examination questions. The overall aim is to provide insights beyond the syllabus to effectively teach the FP3 module on matrices.
This document provides a syllabus for mathematics courses at the secondary and higher secondary levels. Some key points:
- The syllabus builds upon concepts from previous grades in a continuous manner from classes 9 to 12.
- It is designed to be taught in approximately 180 hours to allow sufficient time for exploration and understanding of concepts.
- Areas like proofs and mathematical modeling are introduced gradually from classes 9 to 12 since they are new concepts.
- The syllabus covers topics like number systems, algebra, trigonometry, coordinate geometry, geometry, mensuration, statistics and probability.
- Specific concepts outlined for each class include polynomials, linear equations, quadratic equations, trigonometric ratios and identities.
This document provides an overview of matrices including:
- Definitions of matrices, order of matrices, and compact matrix form
- Matrix multiplication and checking compatibility of matrices
- Determinants, adjoints, and inverses of matrices
- Methods for solving systems of equations using matrices including Gauss-Jordan elimination and Cramer's rule
The document also provides brief biographies of James Joseph Sylvester and Arthur Cayley, two mathematicians who made important contributions to the field of matrices.
The history and development of matrix theory is summarized as follows:
1) The term "matrix" was introduced in 1850 by James Joseph Sylvester to describe rectangular arrays of numbers or expressions arranged in rows and columns.
2) The founder of modern matrix theory is considered to be Arthur Cayley, who in the 1850s introduced concepts such as inverse matrices and matrix multiplication.
3) Important developments in matrix theory continued throughout the 19th and 20th centuries, including the discovery by Arthur Cayley and William Hamilton of unique properties of matrices.
This document defines and describes different types of matrices including:
- Upper and lower triangular matrices
- Determinants which are scalars obtained from products of matrix elements according to constraints
- Band matrices which are sparse matrices with nonzero elements confined to diagonals
- Transpose matrices which exchange the rows and columns of a matrix
- Inverse matrices which when multiplied by the original matrix produce the identity matrix
Thue showed that there exist arbitrarily long square-free strings over an alphabet of three symbols (not true for two symbols). An open problem was posed, which is a generalization of Thue’s original result: given an alphabet list L = L1, . . . , Ln, where |Li| = 3, is it always possible to find a square-free string, w = w1w2 . . . wn, where wi ∈ Li? In this paper we show that squares can be forced on square-free strings over alphabet lists iff a suffix of the square-free string conforms to a pattern which we term as an offending suffix. We also prove properties of offending suffixes. However, the problem remains tantalizingly open.
This talk is going to be centered on two papers that are going to appear in the following months:
Neerja Mhaskar and Michael Soltys, Non-repetitive strings over alphabet lists
to appear in WALCOM, February 2015.
Neerja Mhaskar and Michael Soltys, String Shuffle: Circuits and Graphs
to appear in the Journal of Discrete Algorithms, January 2015.
Visit http://soltys.cs.csuci.edu for more details (these two papers are number 3 and 19 on the page), as well as Python programs that can be used to illustrate the ideas in the papers. We are going to introduce some basic concepts related to computations on string, present some recent results, and propose two open problems.
Tom Pranayanuntana teaches mathematics at NYU-Poly and consistently receives excellent student evaluations. He teaches concepts algebraically, numerically, graphically, and verbally to help students make connections. Real-world examples and technology like graphing calculators are used to keep students engaged. Flexible office hours and genuine concern for student learning have been effective. Teaching is both a science and an art requiring observing student responses and finding different strategies for understanding.
1) The document discusses using matrices to represent transformations of points in 2D and 3D spaces, as well as state transitions in systems modeled by discrete changes in state variables over time.
2) Transformations like scaling, stretching, shearing, reflection and rotation of a basic shape (unit square) are demonstrated through matrix multiplication.
3) Combining multiple transformations and inverses are also explained through examples like shearing and stretching.
4) Transition matrices are introduced to model systems with discrete state changes, demonstrated through examples like wagon distribution between locations over weeks.
This document provides a syllabus for mathematics courses at the secondary and higher secondary levels. Some key points:
- The syllabus builds upon concepts from previous grades in a continuous manner from classes 9 to 12.
- It is designed to be taught in approximately 180 hours to allow sufficient time for exploration and understanding of concepts.
- Areas like proofs and mathematical modeling are introduced gradually from classes 9 to 12 since they are new concepts.
- The syllabus covers topics like number systems, algebra, trigonometry, coordinate geometry, geometry, mensuration, statistics and probability.
- Specific concepts outlined for each class include polynomials, linear equations, quadratic equations, trigonometric ratios and identities.
This document provides an overview of matrices including:
- Definitions of matrices, order of matrices, and compact matrix form
- Matrix multiplication and checking compatibility of matrices
- Determinants, adjoints, and inverses of matrices
- Methods for solving systems of equations using matrices including Gauss-Jordan elimination and Cramer's rule
The document also provides brief biographies of James Joseph Sylvester and Arthur Cayley, two mathematicians who made important contributions to the field of matrices.
The history and development of matrix theory is summarized as follows:
1) The term "matrix" was introduced in 1850 by James Joseph Sylvester to describe rectangular arrays of numbers or expressions arranged in rows and columns.
2) The founder of modern matrix theory is considered to be Arthur Cayley, who in the 1850s introduced concepts such as inverse matrices and matrix multiplication.
3) Important developments in matrix theory continued throughout the 19th and 20th centuries, including the discovery by Arthur Cayley and William Hamilton of unique properties of matrices.
This document defines and describes different types of matrices including:
- Upper and lower triangular matrices
- Determinants which are scalars obtained from products of matrix elements according to constraints
- Band matrices which are sparse matrices with nonzero elements confined to diagonals
- Transpose matrices which exchange the rows and columns of a matrix
- Inverse matrices which when multiplied by the original matrix produce the identity matrix
Thue showed that there exist arbitrarily long square-free strings over an alphabet of three symbols (not true for two symbols). An open problem was posed, which is a generalization of Thue’s original result: given an alphabet list L = L1, . . . , Ln, where |Li| = 3, is it always possible to find a square-free string, w = w1w2 . . . wn, where wi ∈ Li? In this paper we show that squares can be forced on square-free strings over alphabet lists iff a suffix of the square-free string conforms to a pattern which we term as an offending suffix. We also prove properties of offending suffixes. However, the problem remains tantalizingly open.
This talk is going to be centered on two papers that are going to appear in the following months:
Neerja Mhaskar and Michael Soltys, Non-repetitive strings over alphabet lists
to appear in WALCOM, February 2015.
Neerja Mhaskar and Michael Soltys, String Shuffle: Circuits and Graphs
to appear in the Journal of Discrete Algorithms, January 2015.
Visit http://soltys.cs.csuci.edu for more details (these two papers are number 3 and 19 on the page), as well as Python programs that can be used to illustrate the ideas in the papers. We are going to introduce some basic concepts related to computations on string, present some recent results, and propose two open problems.
Tom Pranayanuntana teaches mathematics at NYU-Poly and consistently receives excellent student evaluations. He teaches concepts algebraically, numerically, graphically, and verbally to help students make connections. Real-world examples and technology like graphing calculators are used to keep students engaged. Flexible office hours and genuine concern for student learning have been effective. Teaching is both a science and an art requiring observing student responses and finding different strategies for understanding.
1) The document discusses using matrices to represent transformations of points in 2D and 3D spaces, as well as state transitions in systems modeled by discrete changes in state variables over time.
2) Transformations like scaling, stretching, shearing, reflection and rotation of a basic shape (unit square) are demonstrated through matrix multiplication.
3) Combining multiple transformations and inverses are also explained through examples like shearing and stretching.
4) Transition matrices are introduced to model systems with discrete state changes, demonstrated through examples like wagon distribution between locations over weeks.
The document defines sequences and their different types. It discusses arithmetic sequences which have a common difference, and geometric sequences which have a common ratio. It provides examples of sequences and explains how to find the nth term or general term of a sequence given initial terms. The document also covers calculating means of sequences such as arithmetic, harmonic and geometric means, and finding sums of finite and infinite sequences. Examples are given of problems involving finding terms and sums of sequences.
The document defines various types of matrices including row vectors, column vectors, submatrices, square matrices, triangular matrices, diagonal matrices, identity matrices, zero matrices, and diagonally dominant matrices. It provides examples of each type of matrix. It also discusses when two matrices are considered equal, which is when they have the same size and corresponding elements are equal.
1. The document discusses various types and operations of matrices including transpose, similarity, inverse, and determinant of matrices.
2. It also discusses using matrices to solve systems of linear equations by finding the inverse of the coefficient matrix or calculating the determinant.
3. The key matrix concepts covered are matrix notation, types of matrices, matrix addition/subtraction, multiplication, and using matrices to represent and solve linear systems.
Data Analysis and Algorithms Lecture 1: IntroductionTayyabSattar5
This document outlines a course on design and analysis of algorithms. It covers topics like algorithm complexity analysis using growth functions, classic algorithm problems like the traveling salesperson problem, and algorithm design techniques like divide-and-conquer, greedy algorithms, and dynamic programming. Example algorithms and problems are provided for each topic. Reference books on algorithms are also listed.
Slope, Parallel and Perpendicular Lines
SWBAT verbally tell me how the slope of parallel lines and perpendicular lines are related. They can find the equation of a line parallel/perpendicular to a given line that passes through a given point.
Students will learn to convert graphs into matrices and use the concept of matrix equality to solve simple equations. Specifically, they will understand that a matrix represents the number of roads between towns, with elements indicating roads between locations. They will also learn that two matrices are equal only when their order and all elements are the same, and this can be used to calculate unknown values in a matrix equation.
This document outlines the course details for MA121 Mathematics II. The course is designed to provide concepts of Linear Algebra and an introduction to complex variables. It will cover topics such as complex numbers, functions, derivatives, integrals, Taylor series, matrices, vector spaces, linear transformations, eigenvalues and eigenvectors. Students will be evaluated based on internal exams, a midterm, comprehensive exam, with an emphasis on closed book exams. The course aims to help students develop skills in applying concepts of complex analysis and linear algebra.
Matrix and its operation (addition, subtraction, multiplication)NirnayMukharjee
This document summarizes matrix operations including addition, subtraction, and multiplication. It defines a matrix as a rectangular arrangement of numbers in rows and columns. Matrix addition and subtraction can only be done on matrices with the same dimensions, by adding or subtracting the corresponding elements. Matrix multiplication involves multiplying the rows of the first matrix with the columns of the second matrix and summing the products to form the elements of the resulting matrix. Examples are provided to illustrate each operation.
The document defines sequences and their different types, including arithmetic and geometric sequences. It provides examples of finding the nth term or general term of sequences given initial terms. It also discusses how to find the common difference of arithmetic sequences and common ratio of geometric sequences. The document explains how to calculate arithmetic means, harmonic means, and geometric means of sequences. It provides information on finding the sums of finite and infinite arithmetic, harmonic, and geometric sequences.
This document is an introduction to matrices presented by Reza At-Tanzil of the Department of Pharmacy at Comilla University. It defines what a matrix is, describes different types of matrices including column/row matrices, rectangular matrices, square matrices, diagonal matrices, identity matrices, null matrices, and scalar matrices. It also covers matrix operations such as matrix multiplication, transpose of a matrix, symmetric matrices, inverse of a matrix, and adjoint matrices.
A linear graph uses a straight line to represent the relationship between two quantities, where the straight line is plotted on a coordinate plane with perpendicular x- and y-axes to show how one variable depends on the other. Linear graphs can take various forms depending on whether the line is parallel to the x- or y-axis or intersecting at an angle, and they have important applications in areas like data analysis, health monitoring, and interpreting information in daily life. Formulas are provided to calculate and graph lines in slope-intercept or standard form.
The document discusses different types of number patterns (pola bilangan) in mathematics, including: odd numbers, even numbers, square numbers, rectangular numbers, triangular numbers, arithmetic sequences, geometric sequences, Fibonacci sequences, and Pascal's triangle. It provides examples and formulas for determining the nth term of each pattern. It also gives examples of problems involving identifying values within the different patterns.
Today's math lesson will cover graphing quadratic functions by finding the vertex and axis of symmetry. Students will graph 6 quadratic functions as class work. It is recommended to take good notes and bring a calculator every day. Notebooks will be submitted next week. The document then reviews key aspects of quadratic equations and functions, including their standard forms and how to find the x-intercepts, axis of symmetry, and vertex of a parabola. Students' assignment is to graph equations and pay attention to how changing a, b, and c values affects the parabola shape.
The document is a lesson on identifying linear and nonlinear functions from tables of values or graphs. It includes examples of determining whether functions are linear or nonlinear based on whether the rate of change is constant as the input variable changes. It also provides examples of writing equations in y=mx+b form from linear functions and describing qualitative features of functions from graphs or descriptions.
The document discusses finding equations of lines that are parallel or perpendicular to given lines. It provides the definitions that parallel lines have the same slope, while perpendicular lines have slopes that are negative reciprocals of each other. Examples are worked through using the point-slope formula to find equations of parallel and perpendicular lines passing through given points.
The document discusses matrices and their types and applications. It defines a matrix as a rectangular arrangement of numbers, expressions or symbols arranged in rows and columns. It describes 10 different types of matrices including row, column, square, null, identity, diagonal, scalar, transpose, symmetric and equal matrices. It also discusses three algebraic operations on matrices: addition, subtraction and multiplication. Finally, it provides examples of how matrices are used in economics to calculate costs of production, in geology for seismic surveys, and in robotics and automation to program robot movements.
Matrices can be added, subtracted, and multiplied according to certain rules.
- Matrices can only be added or subtracted if they are the same size. The sum or difference of matrices A and B yields a matrix C of the same size.
- Matrices can be multiplied by a scalar. Multiplying a matrix A by a scalar k results in a new matrix kA where each element is multiplied by k.
- Matrix multiplication allows combining information from two matrices but has specific rules regarding the dimensions of the matrices.
The document defines linear equations in two variables as equations that can be written in the form ax + by + c = 0, where a, b, and c are real numbers and a and b are not equal to 0. It provides an example linear equation as 2x + 3y = 18 and explains how to determine if a given ordered pair (3, 4) is a solution by substituting the values into the equation.
Brief review on matrix Algebra for mathematical economicsfelekephiliphos3
This document provides an overview of matrix algebra concepts including:
- Definitions of matrices, vectors, scalars, and different types of matrices like identity, symmetric, diagonal, and triangular matrices
- Operations that can be performed on matrices like addition, multiplication, and determining the determinant
- Calculating minors and cofactors of a matrix elements and using them to find the adjoint and inverse of a matrix
- The process of transposing a matrix by swapping its rows and columns
The document provides lecture notes for a course on matrix algebra for engineers. It covers topics such as the definition of matrices, addition and multiplication of matrices, special matrices like the identity and zero matrices, transposes, inverses, orthogonal matrices, and systems of linear equations. The notes are intended to teach the basics of matrix algebra at a level appropriate for engineering students who have taken calculus. They include video links, examples, problems at the end of each section, and solutions to the problems in an appendix.
1) A matrix is a rectangular array of numbers arranged in rows and columns. The dimensions of a matrix are specified by the number of rows and columns.
2) The inverse of a square matrix A exists if and only if the determinant of A is not equal to 0. The inverse of A, denoted A^-1, is the matrix that satisfies AA^-1 = A^-1A = I, where I is the identity matrix.
3) For two matrices A and B to be inverses, their product must result in the identity matrix regardless of order, i.e. AB = BA = I. This shows that one matrix undoes the effect of the other.
Matrix and its applications by mohammad imranMohammad Imran
This document provides an overview of matrix mathematics concepts. It discusses how matrices are useful in engineering calculations for storing values, solving systems of equations, and coordinate transformations. The outline then reviews properties of matrices and covers various matrix operations like addition, multiplication, and transposition. It also defines different types of matrices and discusses determining the rank, inverse, eigenvalues and eigenvectors of matrices. Key matrix algebra topics like solving systems of equations and putting matrices in normal form are summarized.
The document defines sequences and their different types. It discusses arithmetic sequences which have a common difference, and geometric sequences which have a common ratio. It provides examples of sequences and explains how to find the nth term or general term of a sequence given initial terms. The document also covers calculating means of sequences such as arithmetic, harmonic and geometric means, and finding sums of finite and infinite sequences. Examples are given of problems involving finding terms and sums of sequences.
The document defines various types of matrices including row vectors, column vectors, submatrices, square matrices, triangular matrices, diagonal matrices, identity matrices, zero matrices, and diagonally dominant matrices. It provides examples of each type of matrix. It also discusses when two matrices are considered equal, which is when they have the same size and corresponding elements are equal.
1. The document discusses various types and operations of matrices including transpose, similarity, inverse, and determinant of matrices.
2. It also discusses using matrices to solve systems of linear equations by finding the inverse of the coefficient matrix or calculating the determinant.
3. The key matrix concepts covered are matrix notation, types of matrices, matrix addition/subtraction, multiplication, and using matrices to represent and solve linear systems.
Data Analysis and Algorithms Lecture 1: IntroductionTayyabSattar5
This document outlines a course on design and analysis of algorithms. It covers topics like algorithm complexity analysis using growth functions, classic algorithm problems like the traveling salesperson problem, and algorithm design techniques like divide-and-conquer, greedy algorithms, and dynamic programming. Example algorithms and problems are provided for each topic. Reference books on algorithms are also listed.
Slope, Parallel and Perpendicular Lines
SWBAT verbally tell me how the slope of parallel lines and perpendicular lines are related. They can find the equation of a line parallel/perpendicular to a given line that passes through a given point.
Students will learn to convert graphs into matrices and use the concept of matrix equality to solve simple equations. Specifically, they will understand that a matrix represents the number of roads between towns, with elements indicating roads between locations. They will also learn that two matrices are equal only when their order and all elements are the same, and this can be used to calculate unknown values in a matrix equation.
This document outlines the course details for MA121 Mathematics II. The course is designed to provide concepts of Linear Algebra and an introduction to complex variables. It will cover topics such as complex numbers, functions, derivatives, integrals, Taylor series, matrices, vector spaces, linear transformations, eigenvalues and eigenvectors. Students will be evaluated based on internal exams, a midterm, comprehensive exam, with an emphasis on closed book exams. The course aims to help students develop skills in applying concepts of complex analysis and linear algebra.
Matrix and its operation (addition, subtraction, multiplication)NirnayMukharjee
This document summarizes matrix operations including addition, subtraction, and multiplication. It defines a matrix as a rectangular arrangement of numbers in rows and columns. Matrix addition and subtraction can only be done on matrices with the same dimensions, by adding or subtracting the corresponding elements. Matrix multiplication involves multiplying the rows of the first matrix with the columns of the second matrix and summing the products to form the elements of the resulting matrix. Examples are provided to illustrate each operation.
The document defines sequences and their different types, including arithmetic and geometric sequences. It provides examples of finding the nth term or general term of sequences given initial terms. It also discusses how to find the common difference of arithmetic sequences and common ratio of geometric sequences. The document explains how to calculate arithmetic means, harmonic means, and geometric means of sequences. It provides information on finding the sums of finite and infinite arithmetic, harmonic, and geometric sequences.
This document is an introduction to matrices presented by Reza At-Tanzil of the Department of Pharmacy at Comilla University. It defines what a matrix is, describes different types of matrices including column/row matrices, rectangular matrices, square matrices, diagonal matrices, identity matrices, null matrices, and scalar matrices. It also covers matrix operations such as matrix multiplication, transpose of a matrix, symmetric matrices, inverse of a matrix, and adjoint matrices.
A linear graph uses a straight line to represent the relationship between two quantities, where the straight line is plotted on a coordinate plane with perpendicular x- and y-axes to show how one variable depends on the other. Linear graphs can take various forms depending on whether the line is parallel to the x- or y-axis or intersecting at an angle, and they have important applications in areas like data analysis, health monitoring, and interpreting information in daily life. Formulas are provided to calculate and graph lines in slope-intercept or standard form.
The document discusses different types of number patterns (pola bilangan) in mathematics, including: odd numbers, even numbers, square numbers, rectangular numbers, triangular numbers, arithmetic sequences, geometric sequences, Fibonacci sequences, and Pascal's triangle. It provides examples and formulas for determining the nth term of each pattern. It also gives examples of problems involving identifying values within the different patterns.
Today's math lesson will cover graphing quadratic functions by finding the vertex and axis of symmetry. Students will graph 6 quadratic functions as class work. It is recommended to take good notes and bring a calculator every day. Notebooks will be submitted next week. The document then reviews key aspects of quadratic equations and functions, including their standard forms and how to find the x-intercepts, axis of symmetry, and vertex of a parabola. Students' assignment is to graph equations and pay attention to how changing a, b, and c values affects the parabola shape.
The document is a lesson on identifying linear and nonlinear functions from tables of values or graphs. It includes examples of determining whether functions are linear or nonlinear based on whether the rate of change is constant as the input variable changes. It also provides examples of writing equations in y=mx+b form from linear functions and describing qualitative features of functions from graphs or descriptions.
The document discusses finding equations of lines that are parallel or perpendicular to given lines. It provides the definitions that parallel lines have the same slope, while perpendicular lines have slopes that are negative reciprocals of each other. Examples are worked through using the point-slope formula to find equations of parallel and perpendicular lines passing through given points.
The document discusses matrices and their types and applications. It defines a matrix as a rectangular arrangement of numbers, expressions or symbols arranged in rows and columns. It describes 10 different types of matrices including row, column, square, null, identity, diagonal, scalar, transpose, symmetric and equal matrices. It also discusses three algebraic operations on matrices: addition, subtraction and multiplication. Finally, it provides examples of how matrices are used in economics to calculate costs of production, in geology for seismic surveys, and in robotics and automation to program robot movements.
Matrices can be added, subtracted, and multiplied according to certain rules.
- Matrices can only be added or subtracted if they are the same size. The sum or difference of matrices A and B yields a matrix C of the same size.
- Matrices can be multiplied by a scalar. Multiplying a matrix A by a scalar k results in a new matrix kA where each element is multiplied by k.
- Matrix multiplication allows combining information from two matrices but has specific rules regarding the dimensions of the matrices.
The document defines linear equations in two variables as equations that can be written in the form ax + by + c = 0, where a, b, and c are real numbers and a and b are not equal to 0. It provides an example linear equation as 2x + 3y = 18 and explains how to determine if a given ordered pair (3, 4) is a solution by substituting the values into the equation.
Brief review on matrix Algebra for mathematical economicsfelekephiliphos3
This document provides an overview of matrix algebra concepts including:
- Definitions of matrices, vectors, scalars, and different types of matrices like identity, symmetric, diagonal, and triangular matrices
- Operations that can be performed on matrices like addition, multiplication, and determining the determinant
- Calculating minors and cofactors of a matrix elements and using them to find the adjoint and inverse of a matrix
- The process of transposing a matrix by swapping its rows and columns
The document provides lecture notes for a course on matrix algebra for engineers. It covers topics such as the definition of matrices, addition and multiplication of matrices, special matrices like the identity and zero matrices, transposes, inverses, orthogonal matrices, and systems of linear equations. The notes are intended to teach the basics of matrix algebra at a level appropriate for engineering students who have taken calculus. They include video links, examples, problems at the end of each section, and solutions to the problems in an appendix.
1) A matrix is a rectangular array of numbers arranged in rows and columns. The dimensions of a matrix are specified by the number of rows and columns.
2) The inverse of a square matrix A exists if and only if the determinant of A is not equal to 0. The inverse of A, denoted A^-1, is the matrix that satisfies AA^-1 = A^-1A = I, where I is the identity matrix.
3) For two matrices A and B to be inverses, their product must result in the identity matrix regardless of order, i.e. AB = BA = I. This shows that one matrix undoes the effect of the other.
Matrix and its applications by mohammad imranMohammad Imran
This document provides an overview of matrix mathematics concepts. It discusses how matrices are useful in engineering calculations for storing values, solving systems of equations, and coordinate transformations. The outline then reviews properties of matrices and covers various matrix operations like addition, multiplication, and transposition. It also defines different types of matrices and discusses determining the rank, inverse, eigenvalues and eigenvectors of matrices. Key matrix algebra topics like solving systems of equations and putting matrices in normal form are summarized.
Chapter 5: Determinant
Covered Topics:
5.1 Definition of Determinant
5.2 Expansion of Determinant of order 2X3
5.3 Crammer’s rule to solve simultaneous equations in 3 unknowns
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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.
This document provides an overview of matrices and basic matrix operations. It discusses what matrices are, how to perform operations like addition, multiplication, and taking the transpose. It also covers special types of matrices like diagonal, triangular, and identity matrices. It explains how to calculate the determinant of a 2x2 matrix and find the inverse of a 2x2 matrix using the determinant. The goal is for the reader to understand matrices, common operations, and how to calculate the determinant and inverse of a 2x2 matrix after reviewing this material.
Phase plane analysis is a graphical technique used to study the behavior of nonlinear dynamical systems. It constructs trajectories in a two-dimensional plane using the states of a second-order system as coordinates. This allows visualization of the system's behavior for different initial conditions without solving the nonlinear equations analytically. Key features include singular points where trajectories intersect, constructing trajectories using analytical or graphical methods like isoclines, and understanding symmetry in the phase plane portrait.
1. This document discusses methods for solving linear algebraic equations and operations involving matrices. It covers topics such as matrix definitions, types of matrices, matrix operations, representing equations in matrix form, and methods for solving systems of linear equations including graphical methods, determinants, Cramer's rule, elimination, Gauss-Jordan, LU decomposition, and calculating the matrix inverse.
2. Key matrix operations include addition, multiplication, and rules for inverting a matrix. Methods for solving systems of equations include graphical techniques, determinants, Cramer's rule, elimination, Gauss, Gauss-Jordan, and LU decomposition.
3. LU decomposition involves writing a matrix as the product of a lower and upper triangular matrix, which can
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.
This document discusses techniques for setting linear algebra problems in a way that ensures relatively easy arithmetic. Some key techniques discussed include:
1. Using Pythagorean triples and sums of squares to generate vectors with integer norms in R2 and R3.
2. Using the PLU decomposition theorem to generate matrices with a given determinant, such as ±1, to avoid fractions.
3. Extending a basis for the kernel of a matrix to generate matrices with a given kernel.
4. Ensuring the coefficients for a Leontieff input-output model are nonnegative to generate a productive consumption matrix. Examples and Maple routines are provided.
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
This document discusses using systems of linear equations and matrices to represent and find the intersection of planes in three-dimensional space. It provides examples of using the inverse matrix method and reduced row echelon form (RREF) method to solve systems of 2 and 3 planes. The RREF method can find lines of intersection even when planes do not intersect at a single point, and can reveal when planes share a common line of intersection.
The document summarizes key concepts from a lecture on multiple regression analysis:
1) The lecture is divided into three segments covering multiple regression, matrix algebra, and estimation of coefficients.
2) Multiple regression allows predicting an outcome variable from multiple predictor variables using a regression equation. Matrix algebra concepts like correlation matrices are used.
3) Estimation of coefficients involves minimizing the sum of squared residuals to find the coefficient values (B) that provide the best fitting regression model. The normal equations are used to calculate the coefficient estimates.
Solving ONE’S interval linear assignment problemIJERA Editor
This document presents a new method called the Matrix Ones Interval Linear Assignment Method (MOILA) for solving assignment problems with interval costs. It begins with definitions of assignment problems and interval analysis concepts. Then it describes the existing Hungarian method and provides an example solved using both Hungarian and MOILA. MOILA involves creating ones in the assignment matrix and making assignments based on the ones. The document outlines algorithms for MOILA as well as extensions to unbalanced and interval assignment problems. It provides an example of applying MOILA to solve a balanced interval assignment problem and compares the solutions to Hungarian. The document introduces MOILA as a systematic alternative to Hungarian for solving a variety of assignment problem types.
This document provides examples of how linear algebra is useful across many domains:
1) Linear algebra can be used to represent and analyze networks and graphs through adjacency matrices.
2) Differential equations describing complex systems like bridges and molecules can be understood through matrix representations and eigenvalues.
3) Quantum computing uses linear algebra operations like matrix multiplication to represent computations on quantum bits.
4) Many other areas like coding/encryption, data compression, solving systems of equations, computer graphics, statistics, games, and neural networks rely on concepts from linear algebra.
Linear equation in one variable PPT.pdfmanojindustry
This document discusses linear equations in two variables. It defines linear equations and explains that a linear equation in two variables can be written as ax + by = c, where a, b, and c are real numbers and a and b are not both equal to zero. It also explains that a linear equation in two variables has infinitely many solutions and that the graph of a linear equation is a straight line. The document provides examples of linear equations and their graphical representations.
The document discusses different methods for minimizing Boolean functions, including algebraic manipulation, tabular methods, and Karnaugh maps. The tabular method involves grouping minterms based on their binary representations and combining terms that differ by one bit. Karnaugh maps provide a visual way to group adjacent minterms and identify prime implicants to find a minimized expression. Both methods aim to cover all minterms with the fewest prime implicants.
Similar to Further pure mathematics 3 matrix algebra (20)
The document describes the Vigenère cipher method of encryption. It involves using a keyword that is repeated above the plaintext message. Each letter of the plaintext is then shifted according to the corresponding letter of the keyword, using a table of letter shifts. The document provides instructions for cracking a Vigenère cipher by finding repeating blocks of letters and using the frequency of letters to deduce the keyword length and decrypt the message.
1. Al-Khwarizmi wrote the first treatise on algebra, Hisab al-jabr w’al-muqabala, in 820 AD, which provided methods for solving equations. The word "algebra" is derived from "al-jabr" in the title, meaning restoration.
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The document discusses conic sections including the ellipse and hyperbola. It provides insights beyond the syllabus, including the historical origins of conic sections from Apollonius of Perga's work and their importance in Kepler's laws of planetary motion. The key properties of ellipses and hyperbolas are defined geometrically using focus, directrix, and eccentricity. Their Cartesian and parametric equations are also derived and explained.
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The document discusses equivalent fractions and provides examples of finding equivalent fractions by adding the same number to both the numerator and denominator. It also covers ordering fractions by finding a common denominator and writing fractions with the same denominator. Examples are given of ordering fractions and finding missing numerators or denominators of equivalent fractions.
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it describes the bony anatomy including the femoral head , acetabulum, labrum . also discusses the capsule , ligaments . muscle that act on the hip joint and the range of motion are outlined. factors affecting hip joint stability and weight transmission through the joint are summarized.
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9
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2. Philosophy of the course
• The course does not aim to present a teaching plan for any of
the FP3 topics. This is left to your professional judgement as a
teacher.
• The course is more about presenting each topic in a way that
provides insights into the topic beyond the syllabus.
• In general the course will present the necessary knowledge and
skills to teach the Edexcel FP3 module effectively.
• Relevant examination problem solving techniques will be
demonstrated.
3. Session structure
• Section 1: Introduction to matrices.
• Section 2: Linear transformations represented by
matrices.
• Section 3: The transpose and inverse of a matrix.
• Section 4: Eigenvalues and eigenvectors.
• Section 5. Symmetric matrices and their
reduction to diagonal form.
• Section 6. Typical examination questions.
5. In mathematics a matrix is a rectangular array of
numbers which represents information (such as routes in
a network or rules to effect a transformation in space).
The array is presented within brackets.
Examples of matrices are:
What is a matrix?
0
1
4
1
2
1
3
0
2
,
1
3
,
5
4
3
2
This has 2 rows and 2
columns. It is called a
2 2 matrix.
This has 2 rows and 1
column: a 2 1 matrix
3 rows and 3 columns: a
3 3 matrix.
The entries of a matrix
are referred to by their
(row, column) position.
This is the (1, 2) entry.
This is the (2, 1) entry.
This is the (3, 3) entry.
6. Consider the simultaneous equations:
Another (matrix) way of representing these equations is:
And similarly for the second row:
Such a matrix representation simplifies the study of simultaneous equations
(especially when the number of variables is larger than 2) and, as we’ll see
shortly, also helps us to represent and study transformations of the plane.
How can matrices arise in the classroom?
32
18
5
4
3
2
y
x
32
5
4
18
3
2
y
x
y
x
y
x
y
x
y
x
3
2
3
2
3
2
:
follows
as
column
with the
matrix
the
of
)
(
row
first
he
multiply t
we
LHS
on the
product
he
multiply t
To
y
x
y
x
5
4
5
4
This following slides were presented in the session on FP1 matrices.
7. Let T a 90o anticlockwise turn in the x – y plane about
the origin.
How can matrices arise in the classroom?
1
0
to
by
mapped
gets
0
1
point
The
T
0
1
to
by
mapped
gets
1
0
point
The
T
to
by
mapped
is
at
vertex
one
with
rectangle
blue
the
Now
T
y
x
y
x
x
y
at
vertex
one
with
rectangle
red
..the
x
y
y
x
y
x
x
y
y
x
T
0
1
1
0
:
means
This
y
x
0
1
1
0
T
under
0
1
of
Image
T
under
1
0
of
Image
8. • The earliest evidence of the use of matrices to study simultaneous equations occurs in
the Jiuzhang suanshu or Nine Chapters on the Mathematical Art . This Chinese text
dates from between 206 BC to 50 AD.
• This set of simultaneous equation problem appears in the Jiuzhang suanshu:
• “There are three types of corn, of which three bundles of the first, two of the second, and
one of the third make 39 measures. Two of the first, three of the second and one of the
third make 34 measures. And one of the first, two of the second and three of the third
make 26 measures. How many measures of corn are contained of one bundle of each
type?”
• In algebraic form this is:
• In the Jiuzhang suanshu this is represented in a equivalent form as:
• The text then proceeds to solve the equations using a method identical to the
Gaussian elimination of the late 17 century.
History of the use of matrices to represent simultaneous equations.
26
3
2
1
34
1
3
2
39
1
2
3
z
y
x
z
y
x
z
y
x
39
34
26
1
1
3
2
3
2
3
2
1
The use of matrices to represent transformations is relatively new. Jan de Witt, a
Dutch mathematician, used them in his 1659 text Elements of Curves.
10. •The study of linear transformation is
essentially an extension of the FP1 topic
now extended to 3 3 real matrices.
However it will be instructive to review
the earlier exposition in the following
slides.
11. The transformation: a 90o anticlockwise rotation about the origin.
y
x
vector
a
Consider
y
x
)
1
.(
..........
:
is
rule
then the
,
write
we
If u
kT
u
k
T
y
x
u
:
to
maps
x
y
y
x
T
x
y
y
x
k
vector
enlarged
he
consider t
Now
y
x
k
:
to
maps
evidently
x
y
k
y
x
k
T
x
y
k
y
x
kT
x
y
k
y
x
k
T
Thus
13. Linear transformations of real 3 dimensional space.
numbers.
real
are
and
i.e.
short)
for
space
3
or
(
dimensions
3
with
space
real
in the
;
vectors
any two
and
scalar
any
Consider
1
1
1
3
1
1
1
z
, y
z, x
y,
x,
z
y
x
v
z
y
x
u
k
v
T
u
T
v
u
T
ii
u
kT
u
k
T
i
T
)
)
:
conditions
two
the
satisfies
that
spaces
3
or
2
of
mation
A transfor
n.
nsformatio
linear tra
a
called
is
n.
nsformatio
linear tra
a
is
origin
about the
space
l
dimensiona
2
of
rotation
ise
anticlockw
90
the
Thus o
14. Representing a linear transformation of 3 space by a 3 3 matrix.
z
y
x
T
z
y
x
T 0
0
0
space.
3
in
any vector
be
Let
plane.
the
of
n
nsformatio
linear tra
a
be
Let
z
y
x
T
z
T
y
x
T 0
0
0
0
0
0
.
1
0
0
,
0
1
0
,
0
0
1
suppose
And
3
3
3
2
2
2
1
1
1
c
b
a
T
c
b
a
T
c
b
a
T
......
3
2
1
3
2
1
3
2
1
z
y
x
c
c
c
b
b
b
a
a
a
z
y
x
T
z
T
y
T
x
T 0
0
0
0
0
0
:
imply
2)
and
1)
Rules
1
0
0
0
1
0
0
0
1
zT
yT
xT
3
3
3
2
2
2
1
1
1
c
b
a
z
c
b
a
y
c
b
a
x
3
2
1
3
2
1
3
2
1
zc
yc
xc
zb
yb
xb
za
ya
xa
.
of
matrix
the
is
3
2
1
3
2
1
3
2
1
T
c
c
c
b
b
b
a
a
a
15. Linear transformation terminology
space.
3
in
any vector
be
Let
plane.
the
of
n
nsformatio
linear tra
a
be
Let
z
y
x
T
T
z
y
x
c
b
a
c
b
a
z
y
x
T under
of
the
is
say that
we
If
image
3
3
3
2
2
2
1
1
1
1
0
0
,
0
1
0
,
0
0
1
If
c
b
a
T
c
b
a
T
c
b
a
T
.
of
matrix
the
of
columns
the
are
1
0
0
,
0
1
0
,
0
0
1
vectors
the
of
images
The
i.e.
of
matrix
the
slide,
previous
in the
seen
as
Then,
3
2
1
3
2
1
3
2
1
T
c
c
c
b
b
b
a
a
a
T
18. The matrix of a linear transformation. Example.
z
x
y
z
y
x
T
:
n
nsformatio
linear tra
the
of
matrix
the
Find
1
1
1
0
1
0
0
0
1
:
c
b
a
T
Solution
;
0
0
1
0
1
0
2
2
2
c
b
a
T
.
1
0
0
1
0
0
3
3
3
c
b
a
T
.
of
matrix
the
is
1
0
0
0
0
1
0
1
0
3
2
1
3
2
1
3
2
1
T
c
c
c
b
b
b
a
a
a
19. The matrix of a linear transformation. Exercise.
z
y
x
z
y
x
T
3
2
:
n
nsformatio
linear tra
the
of
matrix
the
Find
20. The matrix of a linear transformation. Exercise solution
z
y
x
z
y
x
T
3
2
:
n
nsformatio
linear tra
the
of
matrix
the
Find
;
0
0
1
0
0
1
:
1
1
1
c
b
a
T
Solution
;
0
2
0
0
1
0
2
2
2
c
b
a
T
.
3
0
0
1
0
0
3
3
3
c
b
a
T
.
of
matrix
the
is
3
0
0
0
2
0
0
0
1
3
2
1
3
2
1
3
2
1
T
c
c
c
b
b
b
a
a
a
21. •Section 3: The determinant,
transpose and inverse of a
matrix.
22. • This section is again essentially an extension
of the FP1 topic on now the inverse of a 3 3
real matrix. However a new concept of the
transpose of a matrix will be needed to
compute the inverse matrix.
• While the exposition for determinant in the
2 2 is easy to understand that of the 3 3
case is beyond the scope of this course
(essentially being a second year undergraduate
pure mathematics module).
• One may therefore ask the question as to why
this particular section is included in the FP3
module.
23. What is a determinant? The 2 2 case.
T2
d
b
c
a
M
matrix
n with
nsformatio
linear tra
he
Consider t
case.
general
the
to
go
to
need
really
we
this
understand
To
1
1
and
1
0
,
0
1
,
0
0
ices
with vert
square
the
:
square
unit
the
to
to
does
what
examine
will
We
M
0
1
1
0
1
1
0
0
d
b
c
a
M
d
b
c
a
d
b
c
a
d
c
M
b
a
M
M
1
1
:
i.e.
.
1
1
While
;
1
0
:
;
0
1
:
;
0
0
0
0
:
definition
By
b
a
d
c
d
b
c
a
units.
sq.
1
square
unit
the
of
area
The ram.
parallelog
red
the
into
ed
transform
is
square
unit
The
blue.
in
shown
...
area
its
half
g
calculatin
first
by
ram
parallelog
red
the
of
area
the
find
We
shown.
rectangle
orange
with the
triangle
blue
the
be
Circumscri
T3.
of
Area
T2
of
Area
T1
of
Area
rectangle
orange
the
of
Area
triangle
blue
the
of
area
The
T1
T3
ab
b
d
c
a
cd
ad
2
1
)
)(
(
2
1
2
1
2
)
(
2 ab
bc
cd
ab
ad
cd
ad
2
bc
ad
).
(
area
has
ram
parallelog
red
ed
transform
The bc
ad
24. ).
(
area
with
ram
parallelog
a
1into
area
with
square
unit
the
transforms
matrix
n with
nsformatio
linear tra
that the
see
have
We
bc
ad
d
b
c
a
M
).
(
is
matrix
n with
nsformatio
linear tra
the
of
factor
t
enlargemen
area
the
Therefore
bc
ad
d
b
c
a
M
.
matrix
the
of
the
called
is
)
(
number
The
d
b
c
a
M
bc
ad t
determinan
:
tion'
multiplica
cross
'
a
by
matrix
the
from
calculated
is
)
(
t
determinan
The
d
b
c
a
bc
ad
.
)
(
det
write
We bc
ad
M
What is a determinant? The 2 2 case.
25. :
matrix
has
:
ation
transform
the
seen that
have
We
x
y
y
x
T
0
1
1
0
M
0
1
1
0
0
1
1
0
2
MM
M
0
0
1
1
1
0
0
1
0
1
1
0
1
1
0
0
1
0
0
1
matrix.
the
called
is
1
0
0
1
identity
matrices.
for
'1'
a
as
acts
1
0
0
1
because
is
This
1
0
0
1
1
0
0
1
i.e.
d
b
c
a
d
b
c
a
d
b
c
a
inverse.
own
its
is
say that
we
1
0
0
1
because
and
reason
For this M
MM
)
(
1
0
0
1
such that
matrix
a
is
of
inverse
he
say that t
we
,
matrix
general
a
For
1
1
1
A
A
AA
A
A
d
b
c
a
A
The inverse of a 2 2 matrix.
26.
x
y
y
x
T
:
n
nsformatio
linear tra
he
Consider t
.
represents
t
matrix tha
the
of
inverse
the
from
is
determine
y to
easiest wa
the
Generally 1
T
T
1
written
is
:
;
:
if
of
inverse
the
called
is
)
,
(
)
,
(
:
ation
transform
(linear)
The
T
S
y
x
y
x
ST
y
x
y
x
TS
T
y
x
g
y
x
f
y
x
S
.
0
1
1
0
is
inverse
its
and
0
1
1
0
is
represents
t
matrix tha
n the
calculatio
previous
a
From
1
M
M
T
x
y
y
x
y
x
T
0
1
1
0
:
means
This 1
The inverse of a transformation in 2 space.
27. .
is
say
we
exists
If
).
(
1
0
0
1
such that
matrix
a
is
exists,
it
if
,
of
inverse
he
say that t
we
,
matrix
general
a
For
1
1
1
1
singular
-
non
A
A
A
A
AA
A
A
d
b
c
a
A
:
algebra
cumbersome
some
do
to
need
we
determine
To 1
A
:
equation
the
solves
one
then
and
supposes
first
one
Essentialy 1
s
q
r
p
A
.
and
,
,
of
in terms
and
,
,
obtaining
,
1
0
0
1
d
c
b
a
s
r
q
p
s
q
r
p
d
b
c
a
a
b
c
d
bc
ad
s
q
r
p
A
1
that
is
result
The 1
a
b
c
d
A
det
1
d
c
c
a
a
b
c
d
A
A
A
on
Verificati
det
1
: 1
bc
ad
bc
ad
bc
ad 0
0
1
1
0
0
1
Note that if a matrix A has det A = 0 then it cannot have an inverse
and is called singular. If det A ≠ 0 then A is called non-singular.
The rule for computing the inverse of a 2 2 matrix.
28. .
0
det
providing
,
det
1
inverse
the
has 1
bc
ad
A
a
b
c
d
A
A
d
b
c
a
A
.
of
adjugate
the
of
notion
the
introduce
to
need
first
tion we
generalisa
the
enable
To A
.
is
matrix
1
1
this
of
det
the
:
entry
1)
(1,
ough
column thr
and
row
the
Remove
1.
Step
:
follows
as
of
cofactors
matrixof
the
construct
First we
d
d
a
A
Generalising the rule for the 3 3 case: part i)
.
is
matrix
1
1
this
of
det
the
:
entry
2)
(1,
for the
same
the
Do
2.
Step b
b
c
.
is
matrix
1
1
this
of
det
the
:
entry
1)
(2,
for the
Repeat
3.
Step c
c
b
.
is
matrix
1
1
this
of
det
the
:
entry
2)
(2,
for the
Repeat
4.
Step a
a
d
).
short
for
adj
(
of
matrix
adjugate
the
called
is
this
:
:
)
1
(
t
determiman
ing
correspond
with the
of
entries
)
(
the
replace
Lastly
A
A
a
c
b
d
A
m, n n
m
29. )
1
.........(
adj
a
c
b
d
A
d
b
c
a
A
Generalising the rule for the 3 3 case: part ii)
)
2
.........(
det
)
,
(
)
,
(
adj
of
row
first
with the
row
first
the
of
product
scalar
The
A
bc
ad
b
d
c
a
A
A
)
3
.........(
adj
adj
of
transpose
The
entry.
)
(
the
into
entry
)
(
the
making
by
obtained
is
matrix
square
any
of
transpose
The
a
b
c
d
A
A
m
n,
n
m,
T
)
4
.........(
det
1
adj
det
1
that
see
can
we
Now 1
a
b
c
d
A
A
A
A
T
above.
outlined
steps
four
repeat the
:
same
the
is
matrix
3
3
a
of
inverse
the
computing
for
procedure
The
34. The inverse of a transformation in 3 space.
z
y
z
y
x
z
y
x
z
y
x
T
A
A
A
T
z
y
z
y
x
z
y
x
T
2
1
2
3
2
3
2
1
2
1
2
3
2
3
2
1
1
2
1
2
3
2
3
2
1
1
2
1
2
3
2
3
2
1
2
1
2
3
2
3
2
1
0
0
0
0
1
:
is
ation
transform
inverse
Then the
0
0
0
0
1
is
this
And
.
of
inverse
the
find
Next we
0
0
0
0
1
is
This
.
for
matrix
the
need
First we
:
:
ation
transform
the
of
inverse
the
find
want to
we
Suppose
36. Introduction to eigenvectors and eigenvalues.
6
2
3
6
2
3
6
6
2
3
6
2
3
6
the
ing
correspond
an
is
and
the
ing
correspond
an
is
is,
That
.
the
to
ing
correspond
the
are
lines
invariant
on the
origin
the
from
distinct
points
the
And
.
of
the
called
are
and
factors
t
enlargemen
The
12
18
2
3
0
4
9
0
to
ed
transform
is
2
3
And
.
2
3
form
the
of
are
line
on this
Points
.
2
3
is
line
the
,
3
2
case
the
For
12
18
2
3
0
4
9
0
to
ed
transform
is
2
3
And
.
2
3
form
the
of
are
line
on this
Points
.
2
3
is
line
the
,
3
2
case
the
For
3
2
9
4
9
4
providing
line
on the
lie
will
4
9
Now
.
4
9
0
4
9
0
to
ed
transform
is
This
.
0
,
form
the
of
are
line
on this
origin
the
from
distinct
Points
ation.
transform
by this
invariant
left
are
origin
he
through t
lines
which
find
to
aim
We
.
0
4
9
0
is
matrix
whose
space
2
of
n
nsformatio
linear tra
he
Consider t
2
eigenvalue
r
eigenvecto
eigenvalue
r
eigenvecto
s
eigenvalue
rs
eigenvecto
A
s
eigenvalue
a
a
a
a
a
a
a
a
x
y
m
a
a
a
a
a
a
a
a
x
y
m
m
m
m
ma
a
mx
y
a
ma
a
ma
ma
a
a
ma
a
mx
y
A
a
a
a
a
a
a
a
a
37. Algorithm to find to eigenvalues and eigenvectors.
6.
eigenvalue
with
rs
eigenvecto
are
,
0
,
2
3
So
2
3
6
6
4
9
6
0
4
9
0
have
we
6
case
the
For
6.
eigenvalue
with
rs
eigenvecto
are
,
0
,
2
3
So
2
3
6
6
4
9
6
0
4
9
0
have
we
6
case
the
For
.
6
and
6
are
s
eigenvalue
The
0
36
zero
be
must
t
determinan
its
t,
determinan
of
meaning
the
mind
bearing
ly,
Consequent
area.
destroys
so
and
origin
h the
linethroug
particular
a
in
vectors
all
s
annihilate
0
4
9
0
that
means
This
0
0
0
4
9
0
0
0
0
0
0
4
9
0
0
0
0
4
9
0
Then
eigenvalue
with
0
4
9
0
of
r
eigenvecto
an
be
Let
:
follows
as
derived
is
alogrithm
efficient
An
case.
3
3
in the
especially
efficient
not
is
slide
previous
the
of
method
The
2
a
a
a
x
y
y
x
x
y
y
x
y
x
a
a
a
x
y
y
x
x
y
y
x
y
x
y
x
y
x
y
x
y
x
y
x
A
y
x
0000
00000
00000
00000
00000
00000
00000
00000
00000
38. Eigenvalues and eigenvectors: Example 1 in the 3 3 case.
continued
3.
eigenvalue
with
rs
eigenvecto
are
,
0
,
0
0
value
zero
-
non
any
can take
and
0
0
0
0
2
4
4
3
8
3
3
3
3
2
4
7
3
5
3
3
2
4
0
7
3
0
0
5
have
we
3
case
the
For
.
eigenvalue
each
to
ing
correspond
rs
eigenvecto
the
Find
2.
Step
.
5
and
7
3,
are
rs
eigenvecto
0
3
7
5
0
2
4
7
3
det
1
0
3
4
0
3
det
1
0
3
2
0
7
det
1
5
.
the
is
this
:
0
3
2
4
0
7
3
0
0
5
det
Solve
1.
Step
3
2
4
0
7
3
0
0
5
of
rs
eigenvecto
and
s
eigenvalue
the
Find
4
3
2
a
a
z
y
x
y
x
y
x
x
z
y
x
z
y
x
y
x
x
z
y
x
z
y
x
equation
stic
characteri
A
0 0
0 0
0
39. Eigenvalues and eigenvectors: Example 1 in the 3 3 case (contd)
.
a
a
a
a
y
y
y
z
y
x
y
x
z
y
x
z
y
x
y
x
z
y
x
z
y
x
y
x
x
z
y
x
z
y
x
a
a
a
z
y
x
z
y
x
x
x
z
y
x
z
y
x
y
x
x
z
y
x
z
y
x
5
eigenvalue
with
rs
eigenvecto
are
,
0
,
25
.
2
4
25
.
2
4
1
2
and
4
4
1
2
1
and
4
0
0
0
8
2
4
12
3
0
5
5
5
3
2
4
7
3
5
5
3
2
4
0
7
3
0
0
5
have
we
5
case
the
For
7.
eigenvalue
with
rs
eigenvecto
are
,
0
,
2
0
2
and
0
0
0
0
4
2
4
3
12
7
7
7
3
2
4
7
3
5
7
3
2
4
0
7
3
0
0
5
have
we
7
case
the
For
40. Eigenvalues and eigenvectors: Example 2 in the 3 3 case
continued
3.
eigenvalue
with
rs
eigenvecto
are
,
0
,
,
0
5
5
)
3
(
)
2
(
2
0
5
5
)
2
(
3
)
1
(
)
3
........(
)
2
.......(
)
1
.......(
0
0
0
2
3
2
3
2
3
3
3
3
5
3
3
1
1
3
1
5
1
3
1
1
have
we
3
For
.
eigenvalue
each
to
ing
correspond
rs
eigenvecto
the
Find
2.
Step
.
2
and
6
3,
are
rs
eigenvecto
:
0
6
3
2
0
18
9
2
0
36
7
0
42
9
2
4
10
7
0
14
3
3
2
4
6
1
0
1
3
5
1
det
1
3
1
3
1
1
det
1
1
1
1
1
5
det
1
1
0
1
1
3
1
5
1
3
1
1
det
Solve
1.
Step
1
1
3
1
5
1
3
1
1
of
rs
eigenvecto
and
s
eigenvalue
the
Find
2
2
3
2
3
2
4
3
2
a
a
a
a
x
z
x
y
y
x
y
x
z
y
x
z
y
x
z
y
x
z
y
x
z
y
x
z
y
x
z
y
x
z
y
x
z
y
x
equation
stic
characteri
A
0
0
0
0
0
41. Eigenvalues and eigenvectors: Example 2 in the 3 3 case (contd)
continued
2.
eigenvalue
with
rs
eigenvecto
are
,
0
,
0
,
0
0
20
)
3
(
)
2
(
3
0
20
)
2
(
3
)
1
(
)
3
........(
)
2
.......(
)
1
.......(
0
0
0
3
3
7
3
3
2
2
2
3
5
3
2
1
1
3
1
5
1
3
1
1
have
we
2
For
.
6.
eigenvalue
with
rs
eigenvecto
are
,
0
,
2
2
,
0
4
4
)
3
(
)
2
(
0
4
4
)
2
(
)
1
(
)
3
........(
)
2
.......(
)
1
.......(
0
0
0
5
3
3
5
6
6
6
3
5
3
6
1
1
3
1
5
1
3
1
1
have
we
6
For
.
a
a
a
x
z
y
y
y
z
y
x
z
y
x
z
y
x
z
y
x
z
y
x
z
y
x
z
y
x
z
y
x
z
y
x
a
a
a
a
x
y
x
z
z
x
z
x
z
y
x
z
y
x
z
y
x
z
y
x
z
y
x
z
y
x
z
y
x
z
y
x
z
y
x
0 0
0
47. Definition of a symmetric matrix.
matrices.
3
3
symmetric
are
5
4
2
4
1
3
2
3
1
and
0
2
3
2
2
1
3
1
3
So
.
if
symmetric
is
matrix
square
A
B
A
A
A
A T
48. Eigenvalues and eigenvectors of a symmetric matrix.
.)
resp
(
1
0
1
2
;
1
2
1
6
;
1
1
1
3
1
0
1
;
1
2
1
;
1
1
1
1
1
3
1
5
1
3
1
1
2)
products).
scalar
take
:
(proof
other
each
lar to
perpendicu
are
rs
eigenvecto
three
these
1)
:
Note
.)
resp
2
and
6
3,
s
eigenvalue
to
ding
(correspon
1
0
1
;
1
2
1
:
1
1
1
are
rs
eigenvecto
Particular
2.
eigenvalue
with
rs
eigenvecto
are
,
0
,
0
6;
eigenvalue
with
rs
eigenvecto
are
,
0
,
2
3;
eigenvalue
with
rs
eigenvecto
are
,
0
,
:
are
These
rs.
eigenvecto
and
s
eigenvalue
its
computed
slide
previous
a
in
have,
we
and
symmetric
evidently
is
1
1
3
1
5
1
3
1
1
a
a
a
a
a
a
a
a
a
a
a
A
0
0
0
0
49. Eigenvalues and eigenvectors of a symmetric matrix. (contd)
D
AP
P
DP
AP
P
P
DP
AP
D
P
A
1
1
So
exists.
,
0
3
1
1
1
2
1
det
As
then
s
eigenvalue
of
matrix
diagonal
the
as
2
0
0
0
6
0
0
0
3
and
rs
eigenvecto
lar
perpendicu
of
matrix
the
as
1
1
1
0
2
1
1
1
1
write
we
if
So
......(*)
..........
2
0
0
0
6
0
0
0
3
1
1
1
0
2
1
1
1
1
1
1
1
0
2
1
1
1
1
1
1
3
1
5
1
3
1
1
:
writing
as
thesame
is
that this
show
will
arithmetic
Matrix
.)
resp
(
1
0
1
2
;
1
2
1
6
;
1
1
1
3
1
0
1
;
1
2
1
;
1
1
1
1
1
3
1
5
1
3
1
1
satisfy
rs
eigenvecto
and
s
eigenvalue
its
and
symmetric
evidently
is
1
1
3
1
5
1
3
1
1
0
0
0
0
0
0
50. Diagonalising a symmetric matrix 1.
.
determine
to
required
not
is
one
always,
not
but
Generally,
.
called
is
equation
the
writing
and
,
rs,
eigenvecto
s,
eigenvalue
finding
of
process
the
matrix
symmetric
a
Given
.
s,
eigenvalue
of
matrix
diagonal
the
as
2
0
0
0
6
0
0
0
3
rs,
eigenvecto
orthogonal
of
matrix
the
is
1
1
1
0
2
1
1
1
1
where
,
,
1
1
3
1
5
1
3
1
1
for
that
found
We
1
1
1
P
A
D
AP
P
D
P
A
D
P
D
AP
P
A
ing
diagonalis
matrix.
symmetric
every
to
applies
This
in
s
eigenvalue
of
order
the
to
correspond
in
rs
eigenvecto
the
of
order
the
and
D
P
0
0
0
0
0
0
51. Diagonalising a symmetric matrix 2.
matrix.
orthogonal
1.
length
of
rs
eigenvecto
an
called
is
then
1
length
have
rs
eigenvecto
the
If
find
to
need
that we
is
process
ing
diagonalis
in this
difference
The
now
is
equation
ing
diagonalis
the
and
so
1
0
0
0
1
0
0
0
1
2
1
0
2
1
6
1
6
2
6
1
3
1
3
1
3
1
2
1
6
1
3
1
0
6
2
3
1
2
1
6
1
3
1
now
However
s
eigenvalue
of
matrix
diagonal
the
as
2
0
0
0
6
0
0
0
3
where
,
have
still
we
2
1
6
1
3
1
0
6
2
3
1
2
1
6
1
3
1
then
1
length
of
rs
eigenvecto
consider
now
we
If
.
1
1
3
1
5
1
3
1
1
for
ion
considerat
Another
1
1
P
D
AP
P
P
P
PP
D
D
AP
P
P
A
T
T
T
0
0
0
0
0
0
0
55.
0
1
9
1
0
2
1
k
k
A
(a) Find values of k for which A is singular.
Given that A is non-singular
(b) Find, in terms of k, A–1
Example 2.
k
k
k
k
k
k
k
k
A
A
A
k
k
k
k
k
k
k
k
k
k
k
k
k
k
A
k
k
A
k
k
A
k
k
k
k
k
k
k
A
Solution
T
9
9
18
9
2
2
6
3
1
adj
det
1
2
9
18
2
9
9
1
0
1
det
0
2
det
1
2
1
det
1
9
1
det
0
9
2
det
0
1
2
1
det
1
9
1
0
det
0
9
0
det
0
1
1
det
adj
(b)
6.
or
3
hen
singular w
is
6
or
3
when
0
det
6
3
18
9
1
9
1
0
det
2
0
9
0
det
0
1
1
det
det
(a)
2
1
2
2
0
0
0
0 0
0
56.
3
4
4
4
5
0
4
0
1
A
(a) Verify that is an eigenvector of A and find the corresponding
eigenvalue.
(b) Show that 9 is another eigenvalue of A and find the corresponding
eigenvector.
(c) Find another eigenvalue and corresponding eigenvector and write down a
matrix P and a diagonal matrix D such that P–1 AP = D.
Example 3.
1
2
2
.
9
and
3
,
3
are
rs
eigenvecto
0
9
3
3
0
27
6
3
3
factor
a
is
there
(a)
part
:
0
81
9
9
0
16
80
1
7
9
0
5
4
4
0
1
8
1
0
3
4
4
4
5
0
4
0
1
det
is
The
(b)
3.
eigenvalue
with
of
r
eigenvecto
an
is
1
2
2
1
2
2
3
3
6
6
1
2
2
3
4
4
4
5
0
4
0
1
1
2
2
(a)
2
2
3
2
3
2
ion
stic equat
characteri
A
A
Soln. 0 0
0
0
0
0
57.
3
4
4
4
5
0
4
0
1
A
(b) Show that 9 is another eigenvalue of A and find the corresponding
eigenvector.
(c) Find another eigenvalue and corresponding eigenvector and write down a
matrix P and a diagonal matrix D such that P–1 AP = D.
Example 3 (contd)
3
0
0
0
3
0
0
0
9
1
1
1
2
1
1
2
1
2
so
3,
eigenvalue
with
of
r
eigenvecto
an
is
1
2
2
Also
.
3
eigenvalue
with
of
r
eigenvecto
an
is
1
1
1
9;
eigenvalue
with
of
r
eigenvecto
an
is
1
1
2
.
;
2
0
0
0
4
4
4
2
4
2
;
6
4
4
4
4
4
8
3
3
3
;
9
9
9
3
4
4
4
5
4
3
;
9
3
4
4
4
5
0
4
0
1
.
3
and
9
to
ing
correspond
rs
eigenvecto
of
n
Calculatio
(c)
,
(b)
1
AP
P
P
A
A
A
z
y
z
x
z
y
x
z
y
x
z
y
z
x
z
y
x
z
y
z
x
z
y
x
z
y
x
z
y
x
z
y
z
x
z
y
x
z
y
x
z
y
x
Soln.
0
0 0
0
0
58. 0.
>
and
constants
are
and
where
.
0
3
1
4
1
a
c
p, a, b
c
b
a
p
A
Given AAT = kI , for some constant k.
(a) Find the values of p, k, a, b and c.
(b) Find det A
Example 4.
2
57
2
6
1
2
12
4
2
3
1
det
2
2
2
2
2
3
0
3
1
4
1
So
2
,
2
2
,
2
2
8
,
,
2
,
18
25
.
0
,
2
,
18
,
4
,
0
3
3
,
3
,
18
0
0
0
0
0
0
3
4
3
9
3
4
3
18
0
0
0
0
0
0
3
4
3
9
3
4
3
18
.
1
0
4
3
1
0
3
1
4
1
:
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
A
A
b
c
a
a
a
b
c
a
a
a
a
a
b
c
a
c
b
a
c
b
a
c
a
p
k
k
k
k
c
b
a
pc
a
c
b
a
pc
a
p
p
c
b
a
p
k
k
k
AA
c
b
a
pc
a
c
b
a
pc
a
p
p
c
b
a
p
c
p
b
a
c
b
a
p
AA
Solution
T
T 0
0 0
0
0 0
0
0 0