Eigen values and eigen vectors in matrices and calculus are explained in a simple and understandable manner for all college students who are studying engineering first year
Here are the steps to solve this problem:
1) Find the characteristic polynomial of A: |A - λI| = 0
= (2 - λ)(-1) - (-1)(−1)
= λ^2 - 4λ + 3
2) Set the characteristic polynomial equal to 0 and solve for the eigenvalues:
λ^2 - 4λ + 3 = 0
(λ - 3)(λ - 1) = 0
Eigenvalues are λ1 = 3, λ2 = 1
3) Find the eigenvectors by solving (A - λI)x = 0 for each eigenvalue:
For λ1 = 3: (A - 3I)x
This document provides an introduction and overview of linear equations. It defines key terms like equations, variables, and solutions. It explains that the goal in solving equations is to find the value of the unknown that makes the statement true. The document outlines various properties of equality that can be used to solve equations, such as distributing the same operation to both sides. It also distinguishes between linear and nonlinear equations. Several examples are provided to demonstrate how to solve different types of linear equations, including those with fractions and those that simplify to linear form. The document also briefly introduces solving power equations, which involve variables raised to powers, as well as equations with fractional exponents.
4. Linear Algebra for Machine Learning: Eigenvalues, Eigenvectors and Diagona...Ceni 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 fourth part which is discussing eigenvalues, eigenvectors and diagonalization.
Here is the link of the first part which was discussing linear systems: https://www.slideshare.net/CeniBabaogluPhDinMat/linear-algebra-for-machine-learning-linear-systems/1
Here are the slides of the second part which was discussing basis and dimension:
https://www.slideshare.net/CeniBabaogluPhDinMat/2-linear-algebra-for-machine-learning-basis-and-dimension
Here are the slides of the third part which is discussing factorization and linear transformations.
https://www.slideshare.net/CeniBabaogluPhDinMat/3-linear-algebra-for-machine-learning-factorization-and-linear-transformations-130813437
The document discusses eigenvalue problems and algorithms for solving them. Eigenvalue problems involve finding the eigenvalues and eigenvectors of a matrix and occur across science and engineering. The properties of the eigenvalue problem, like whether the matrix is real or complex, affect the choice of algorithm. The Power Method is described as an iterative technique for determining the dominant eigenvalue and eigenvector of a matrix. It works by successively applying the matrix to a starting vector to isolate the component in the direction of the dominant eigenvector. Variants can find other eigenvalues like the smallest. General projection methods approximate eigenvectors within a subspace, while subspace iteration generalizes Power Method to compute multiple eigenvalues.
This document discusses two applications of matrices: 1) Solving systems of linear equations by manipulating matrices and 2) Finding eigenvalues and eigenvectors by solving the characteristic equation of a matrix. It provides an example of using matrices to solve a system of 3 equations with 3 unknowns. It also gives an example of finding the eigenvalues (-1, -2) and eigenvectors of a 2x2 matrix and using MatLab functions to calculate them.
Eigen values and Eigen vectors ppt worldraykoustav145
The document provides information about eigenvalues and eigenvectors. It defines them as special scalar values and vectors that are preserved by a linear transformation represented by a matrix. The summary explains that eigenvalues scale eigenvectors and that solving the characteristic equation of a matrix yields its eigenvalues, while corresponding eigenvectors are found by solving systems of equations involving the eigenvalues. It concludes that eigenvalues and eigenvectors are fundamental concepts in linear algebra with applications across many fields.
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.
The document discusses image alignment techniques in computer vision. It covers:
1) Computing transformations between images using matched points, by finding the transformation that minimizes error according to the least squares criterion. This can solve for translations, affine transformations, and homographies.
2) Solving the least squares problem results in a system of linear or linearized equations that can be solved efficiently.
3) Homographies are more complex than translations or affine transforms, as the equations are nonlinear. The problem can still be solved using least squares by taking the eigenvector of the smallest eigenvalue.
Here are the steps to solve this problem:
1) Find the characteristic polynomial of A: |A - λI| = 0
= (2 - λ)(-1) - (-1)(−1)
= λ^2 - 4λ + 3
2) Set the characteristic polynomial equal to 0 and solve for the eigenvalues:
λ^2 - 4λ + 3 = 0
(λ - 3)(λ - 1) = 0
Eigenvalues are λ1 = 3, λ2 = 1
3) Find the eigenvectors by solving (A - λI)x = 0 for each eigenvalue:
For λ1 = 3: (A - 3I)x
This document provides an introduction and overview of linear equations. It defines key terms like equations, variables, and solutions. It explains that the goal in solving equations is to find the value of the unknown that makes the statement true. The document outlines various properties of equality that can be used to solve equations, such as distributing the same operation to both sides. It also distinguishes between linear and nonlinear equations. Several examples are provided to demonstrate how to solve different types of linear equations, including those with fractions and those that simplify to linear form. The document also briefly introduces solving power equations, which involve variables raised to powers, as well as equations with fractional exponents.
4. Linear Algebra for Machine Learning: Eigenvalues, Eigenvectors and Diagona...Ceni 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 fourth part which is discussing eigenvalues, eigenvectors and diagonalization.
Here is the link of the first part which was discussing linear systems: https://www.slideshare.net/CeniBabaogluPhDinMat/linear-algebra-for-machine-learning-linear-systems/1
Here are the slides of the second part which was discussing basis and dimension:
https://www.slideshare.net/CeniBabaogluPhDinMat/2-linear-algebra-for-machine-learning-basis-and-dimension
Here are the slides of the third part which is discussing factorization and linear transformations.
https://www.slideshare.net/CeniBabaogluPhDinMat/3-linear-algebra-for-machine-learning-factorization-and-linear-transformations-130813437
The document discusses eigenvalue problems and algorithms for solving them. Eigenvalue problems involve finding the eigenvalues and eigenvectors of a matrix and occur across science and engineering. The properties of the eigenvalue problem, like whether the matrix is real or complex, affect the choice of algorithm. The Power Method is described as an iterative technique for determining the dominant eigenvalue and eigenvector of a matrix. It works by successively applying the matrix to a starting vector to isolate the component in the direction of the dominant eigenvector. Variants can find other eigenvalues like the smallest. General projection methods approximate eigenvectors within a subspace, while subspace iteration generalizes Power Method to compute multiple eigenvalues.
This document discusses two applications of matrices: 1) Solving systems of linear equations by manipulating matrices and 2) Finding eigenvalues and eigenvectors by solving the characteristic equation of a matrix. It provides an example of using matrices to solve a system of 3 equations with 3 unknowns. It also gives an example of finding the eigenvalues (-1, -2) and eigenvectors of a 2x2 matrix and using MatLab functions to calculate them.
Eigen values and Eigen vectors ppt worldraykoustav145
The document provides information about eigenvalues and eigenvectors. It defines them as special scalar values and vectors that are preserved by a linear transformation represented by a matrix. The summary explains that eigenvalues scale eigenvectors and that solving the characteristic equation of a matrix yields its eigenvalues, while corresponding eigenvectors are found by solving systems of equations involving the eigenvalues. It concludes that eigenvalues and eigenvectors are fundamental concepts in linear algebra with applications across many fields.
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.
The document discusses image alignment techniques in computer vision. It covers:
1) Computing transformations between images using matched points, by finding the transformation that minimizes error according to the least squares criterion. This can solve for translations, affine transformations, and homographies.
2) Solving the least squares problem results in a system of linear or linearized equations that can be solved efficiently.
3) Homographies are more complex than translations or affine transforms, as the equations are nonlinear. The problem can still be solved using least squares by taking the eigenvector of the smallest eigenvalue.
This document describes an undergraduate research project on iterative methods for computing eigenvalues and eigenvectors of matrices. It introduces the standard eigenvalue problem and defines key terms like eigenvalues, eigenvectors, and dominant eigenpairs. The body of the document reviews three iterative methods - the power method, inverse power method, and shifted inverse power method. It explains how these methods use repeated matrix-vector multiplications to approximate dominant, smallest, and intermediate eigenvalues and their corresponding eigenvectors. The document is structured with chapters on introduction, literature review, applications, and conclusion.
This document discusses various matrix decomposition techniques including least squares, eigendecomposition, and singular value decomposition. It begins with an introduction to the importance of linear algebra and decompositions for applications. Then it provides examples of using least squares to fit curves to data and find regression lines. It defines eigenvalues and eigenvectors and provides examples of eigendecomposition. It also discusses diagonalization of matrices and using the eigendecomposition to raise matrices to powers. Finally, it discusses singular value decomposition and its applications.
This document provides a review for an upcoming math test covering several topics:
- Review of addition, subtraction, equations, and an introduction to complex fractions
- Sample problems are provided to review solving quadratic equations algebraically and graphically, as well as rational equations
- Instructions are given for tomorrow's test and how current grades will be calculated, with a proposal that students can pass the quarter by scoring a minimum of 75% on the remaining tests.
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.
This document discusses eigenvectors and eigenvalues. It defines eigenvectors as non-zero vectors that satisfy the equation AX = λX, where λ is the eigenvalue. Properties of eigenpairs are described, such as how eigenvectors can be scaled and how eigenvalues relate to the determinant and trace of the matrix. Methods for finding eigenpairs are presented, including solving the characteristic equation. Applications in areas like Google's PageRank algorithm are also mentioned.
The document discusses the history and concepts of radicals. It explains that Pythagoras and his followers believed that natural numbers and proportions between natural numbers governed the universe. However, the Pythagorean theorem disproved this by showing the existence of irrational numbers like the square root of 2. The key points are:
- Pythagoras' philosophical theory was disproved by the existence of irrational numbers like the square root of 2 from the Pythagorean theorem.
- The Pythagorean theorem states that the square of the hypotenuse of a right triangle equals the sum of the squares of the other two sides.
- Radicals can be used to express solutions to equations and powers with fractional
I am Duncan V. I am a Single Variable Calculus Assignment Solver at mathhomeworksolver.com. I hold a Master's in Mathematics, from Manchester, United Kingdom. I have been helping students with their assignments for the past 12 years. I solved assignments related to Single Variable Calculus.
Visit mathhomeworksolver.com or email support@mathhomeworksolver.com. You can also call on +1 678 648 4277 for any assistance with Single Variable Calculus Assignment.
1. The document defines logarithmic functions and their relationship to exponential functions. Logarithmic functions are the inverse of exponential functions with the same base.
2. Methods for determining the logarithm of a number are presented, including using a logarithm table to find the logarithm of numbers that are not perfect powers of their base.
3. The laws of logarithms are proved using properties of exponents, including laws such as log(ab) = log(a) + log(b) and log(a/b) = log(a) - log(b). Examples of applying the laws of logarithms to equations are provided.
This document provides an overview of different types of equations and inequalities in mathematics, including:
1. Linear equations which contain variables with an exponent of 1 and have one solution. The general steps for solving linear equations are expanding brackets, rearranging terms, and finding the solution.
2. Quadratic equations which contain variables with an exponent of up to 2 and have at most two solutions. The general steps for solving quadratic equations involve rewriting the equation in standard form, factorizing, and finding the solutions.
3. Simultaneous equations which involve solving two equations with two unknown variables simultaneously using substitution or elimination methods to eliminate one variable and solve for the other.
4. Word problems
Optimum Engineering Design - Day 2b. Classical Optimization methodsSantiagoGarridoBulln
This document provides an overview of an optimization methods course, including its objectives, prerequisites, and materials. The course covers topics such as linear programming, nonlinear programming, and mixed integer programming problems. It also includes mathematical preliminaries on topics like convex sets and functions, gradients, Hessians, and Taylor series expansions. Methods for solving systems of linear equations and examples are presented.
The document provides information on solving the sum of subsets problem using backtracking. It discusses two formulations - one where solutions are represented by tuples indicating which numbers are included, and another where each position indicates if the corresponding number is included or not. It shows the state space tree that represents all possible solutions for each formulation. The tree is traversed depth-first to find all solutions where the sum of the included numbers equals the target sum. Pruning techniques are used to avoid exploring non-promising paths.
This document summarizes key concepts from a lecture on linear algebra:
1) It defines terms like linear combinations, linear independence, orthonormal vectors, eigenvalues, and eigendecomposition as they relate to vectors and matrices.
2) It describes how to solve least squares problems by exploiting properties of positive semidefinite matrices like their eigendecomposition.
3) Solving a least squares problem for a positive semidefinite matrix M can be reduced to solving a simpler problem involving its eigenvectors and eigenvalues.
Pres Absolute Value Inequalities (Section 1.8)schrockb
This document discusses rules and properties for solving absolute value equations and inequalities. It states that if a is greater than 0 and the absolute value of x is equal to a, then x equals a or negative a. If the absolute value of x is less than a, x is between negative a and a. If the absolute value of x is greater than a, x is less than negative a or greater than a. It provides examples of solving different types of absolute value equations and inequalities.
Eigen values and eigen vectors engineeringshubham211
mathematics...for engineering mathematics.....learn maths...............................The individual items in a matrix are called its elements or entries.[4] Provided that they are the same size (have the same number of rows and the same number of columns), two matrices can be added or subtracted element by element. The rule for matrix multiplication, however, is that two matrices can be multiplied only when the number of columns in the first equals the number of rows in the second. Any matrix can be multiplied element-wise by a scalar from its associated field. A major application of matrices is to represent linear transformations, that is, generalizations of linear functions such as f(x) = 4x. For example, the rotation of vectors in three dimensional space is a linear transformation which can be represented by a rotation matrix R: if v is a column vector (a matrix with only one column) describing the position of a point in space, the product Rv is a column vector describing the position of that point after a rotation. The product of two transformation matrices is a matrix that represents the composition of two linear transformations. Another application of matrices is in the solution of systems of linear equations. If the matrix is square, it is possible to deduce some of its properties by computing its determinant. For example, a square matrix has an inverse if and only if its determinant is not zero. Insight into the geometry of a linear transformation is obtainable (along with other information) from the matrix's eigenvalues and eigenvectors.
Applications of matrices are found in most scientific fields. In every branch of physics, including classical mechanics, optics, electromagnetism, quantum mechanics, and quantum electrodynamics, they are used to study physical phenomena, such as the motion of rigid bodies. In computer graphics, they are used to project a 3-dimensional image onto a 2-dimensional screen. In probability theory and statistics, stochastic matrices are used to describe sets of probabilities; for instance, they are used within the PageRank algorithm that ranks the pages in a Google search.[5] Matrix calculus generalizes classical analytical notions such as derivatives and exponentials to higher dimensions.
A major branch of numerical analysis is devoted to the development of efficient algorithms for matrix computations, a subject that is centuries old and is today an expanding area of research. Matrix decomposition methods simplify computations, both theoretically and practically. Algorithms that are tailored to particular matrix structures, such as sparse matrices and near-diagonal matrices, expedite computations in finite element method and other computations. Infinite matrices occur in planetary theory and in atomic theory. A simple example of an infinite matrix is the matrix representing the derivative operator, which acts on the Taylor series of a function
...
Antiderivatives, Differential Equations, and Slope Fields.pptKarenGardose
This document discusses antiderivatives, differential equations, and slope fields. It defines antiderivatives as the inverse operation of differentiation, where finding an antiderivative involves taking the integral of a function to find a potential function whose derivative equals the original function. It also explains how to solve basic differential equations by isolating the differential and taking the antiderivative of both sides. Finally, it introduces slope fields as a way to visualize the general behavior of solutions to a differential equation by plotting the slope of the tangent line at various points in the plane.
Antiderivatives, Differential Equations, and Slope Fields (1).pptLaibaRao4
This document discusses antiderivatives, differential equations, and slope fields. It defines antiderivatives as the inverse operation of differentiation, and shows examples of finding antiderivatives. It explains that solving a differential equation involves isolating the differential and taking the antiderivative to find the indefinite solution. The document also defines slope fields as a way to visualize the general behavior of a differential equation's solutions by plotting the slope of the tangent line at different points in the plane.
This document provides information about solving absolute value equations and inequalities, as well as quadratic equations. It discusses:
1) To solve absolute value equations, you must divide the equation into two equations by treating the expression inside the absolute value bars as both positive and negative.
2) For inequalities, the direction of the inequality sign must be reversed when multiplying or dividing both sides by a negative number.
3) Quadratic equations can be solved by factoring if possible, or using the quadratic formula. The discriminant determines the number of real roots.
I am Harvey L. I am a Computation and System Biology Assignment Expert at nursingassignmenthelp.com. I hold a Ph.D. in Biology, from Bond University, Australia. I have been helping students with their assignments for the past 14 years. I solve assignments related to Computation and System Biology.
Visit nursingassignmenthelp.com or email info@nursingassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Computation and System Biology Assignments.
I am Arcady N. I am a Single Variable Calculus Assignment Expert at mathsassignmenthelp.com. I hold a Master's in Mathematics from, Queen’s University. I have been helping students with their assignments for the past 12 years. I solve assignments related to Single Variable Calculus.
Visit mathsassignmenthelp.com or email info@mathsassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Single Variable Calculus Assignment.
How to Build a Module in Odoo 17 Using the Scaffold MethodCeline George
Odoo provides an option for creating a module by using a single line command. By using this command the user can make a whole structure of a module. It is very easy for a beginner to make a module. There is no need to make each file manually. This slide will show how to create a module using the scaffold method.
This document describes an undergraduate research project on iterative methods for computing eigenvalues and eigenvectors of matrices. It introduces the standard eigenvalue problem and defines key terms like eigenvalues, eigenvectors, and dominant eigenpairs. The body of the document reviews three iterative methods - the power method, inverse power method, and shifted inverse power method. It explains how these methods use repeated matrix-vector multiplications to approximate dominant, smallest, and intermediate eigenvalues and their corresponding eigenvectors. The document is structured with chapters on introduction, literature review, applications, and conclusion.
This document discusses various matrix decomposition techniques including least squares, eigendecomposition, and singular value decomposition. It begins with an introduction to the importance of linear algebra and decompositions for applications. Then it provides examples of using least squares to fit curves to data and find regression lines. It defines eigenvalues and eigenvectors and provides examples of eigendecomposition. It also discusses diagonalization of matrices and using the eigendecomposition to raise matrices to powers. Finally, it discusses singular value decomposition and its applications.
This document provides a review for an upcoming math test covering several topics:
- Review of addition, subtraction, equations, and an introduction to complex fractions
- Sample problems are provided to review solving quadratic equations algebraically and graphically, as well as rational equations
- Instructions are given for tomorrow's test and how current grades will be calculated, with a proposal that students can pass the quarter by scoring a minimum of 75% on the remaining tests.
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.
This document discusses eigenvectors and eigenvalues. It defines eigenvectors as non-zero vectors that satisfy the equation AX = λX, where λ is the eigenvalue. Properties of eigenpairs are described, such as how eigenvectors can be scaled and how eigenvalues relate to the determinant and trace of the matrix. Methods for finding eigenpairs are presented, including solving the characteristic equation. Applications in areas like Google's PageRank algorithm are also mentioned.
The document discusses the history and concepts of radicals. It explains that Pythagoras and his followers believed that natural numbers and proportions between natural numbers governed the universe. However, the Pythagorean theorem disproved this by showing the existence of irrational numbers like the square root of 2. The key points are:
- Pythagoras' philosophical theory was disproved by the existence of irrational numbers like the square root of 2 from the Pythagorean theorem.
- The Pythagorean theorem states that the square of the hypotenuse of a right triangle equals the sum of the squares of the other two sides.
- Radicals can be used to express solutions to equations and powers with fractional
I am Duncan V. I am a Single Variable Calculus Assignment Solver at mathhomeworksolver.com. I hold a Master's in Mathematics, from Manchester, United Kingdom. I have been helping students with their assignments for the past 12 years. I solved assignments related to Single Variable Calculus.
Visit mathhomeworksolver.com or email support@mathhomeworksolver.com. You can also call on +1 678 648 4277 for any assistance with Single Variable Calculus Assignment.
1. The document defines logarithmic functions and their relationship to exponential functions. Logarithmic functions are the inverse of exponential functions with the same base.
2. Methods for determining the logarithm of a number are presented, including using a logarithm table to find the logarithm of numbers that are not perfect powers of their base.
3. The laws of logarithms are proved using properties of exponents, including laws such as log(ab) = log(a) + log(b) and log(a/b) = log(a) - log(b). Examples of applying the laws of logarithms to equations are provided.
This document provides an overview of different types of equations and inequalities in mathematics, including:
1. Linear equations which contain variables with an exponent of 1 and have one solution. The general steps for solving linear equations are expanding brackets, rearranging terms, and finding the solution.
2. Quadratic equations which contain variables with an exponent of up to 2 and have at most two solutions. The general steps for solving quadratic equations involve rewriting the equation in standard form, factorizing, and finding the solutions.
3. Simultaneous equations which involve solving two equations with two unknown variables simultaneously using substitution or elimination methods to eliminate one variable and solve for the other.
4. Word problems
Optimum Engineering Design - Day 2b. Classical Optimization methodsSantiagoGarridoBulln
This document provides an overview of an optimization methods course, including its objectives, prerequisites, and materials. The course covers topics such as linear programming, nonlinear programming, and mixed integer programming problems. It also includes mathematical preliminaries on topics like convex sets and functions, gradients, Hessians, and Taylor series expansions. Methods for solving systems of linear equations and examples are presented.
The document provides information on solving the sum of subsets problem using backtracking. It discusses two formulations - one where solutions are represented by tuples indicating which numbers are included, and another where each position indicates if the corresponding number is included or not. It shows the state space tree that represents all possible solutions for each formulation. The tree is traversed depth-first to find all solutions where the sum of the included numbers equals the target sum. Pruning techniques are used to avoid exploring non-promising paths.
This document summarizes key concepts from a lecture on linear algebra:
1) It defines terms like linear combinations, linear independence, orthonormal vectors, eigenvalues, and eigendecomposition as they relate to vectors and matrices.
2) It describes how to solve least squares problems by exploiting properties of positive semidefinite matrices like their eigendecomposition.
3) Solving a least squares problem for a positive semidefinite matrix M can be reduced to solving a simpler problem involving its eigenvectors and eigenvalues.
Pres Absolute Value Inequalities (Section 1.8)schrockb
This document discusses rules and properties for solving absolute value equations and inequalities. It states that if a is greater than 0 and the absolute value of x is equal to a, then x equals a or negative a. If the absolute value of x is less than a, x is between negative a and a. If the absolute value of x is greater than a, x is less than negative a or greater than a. It provides examples of solving different types of absolute value equations and inequalities.
Eigen values and eigen vectors engineeringshubham211
mathematics...for engineering mathematics.....learn maths...............................The individual items in a matrix are called its elements or entries.[4] Provided that they are the same size (have the same number of rows and the same number of columns), two matrices can be added or subtracted element by element. The rule for matrix multiplication, however, is that two matrices can be multiplied only when the number of columns in the first equals the number of rows in the second. Any matrix can be multiplied element-wise by a scalar from its associated field. A major application of matrices is to represent linear transformations, that is, generalizations of linear functions such as f(x) = 4x. For example, the rotation of vectors in three dimensional space is a linear transformation which can be represented by a rotation matrix R: if v is a column vector (a matrix with only one column) describing the position of a point in space, the product Rv is a column vector describing the position of that point after a rotation. The product of two transformation matrices is a matrix that represents the composition of two linear transformations. Another application of matrices is in the solution of systems of linear equations. If the matrix is square, it is possible to deduce some of its properties by computing its determinant. For example, a square matrix has an inverse if and only if its determinant is not zero. Insight into the geometry of a linear transformation is obtainable (along with other information) from the matrix's eigenvalues and eigenvectors.
Applications of matrices are found in most scientific fields. In every branch of physics, including classical mechanics, optics, electromagnetism, quantum mechanics, and quantum electrodynamics, they are used to study physical phenomena, such as the motion of rigid bodies. In computer graphics, they are used to project a 3-dimensional image onto a 2-dimensional screen. In probability theory and statistics, stochastic matrices are used to describe sets of probabilities; for instance, they are used within the PageRank algorithm that ranks the pages in a Google search.[5] Matrix calculus generalizes classical analytical notions such as derivatives and exponentials to higher dimensions.
A major branch of numerical analysis is devoted to the development of efficient algorithms for matrix computations, a subject that is centuries old and is today an expanding area of research. Matrix decomposition methods simplify computations, both theoretically and practically. Algorithms that are tailored to particular matrix structures, such as sparse matrices and near-diagonal matrices, expedite computations in finite element method and other computations. Infinite matrices occur in planetary theory and in atomic theory. A simple example of an infinite matrix is the matrix representing the derivative operator, which acts on the Taylor series of a function
...
Antiderivatives, Differential Equations, and Slope Fields.pptKarenGardose
This document discusses antiderivatives, differential equations, and slope fields. It defines antiderivatives as the inverse operation of differentiation, where finding an antiderivative involves taking the integral of a function to find a potential function whose derivative equals the original function. It also explains how to solve basic differential equations by isolating the differential and taking the antiderivative of both sides. Finally, it introduces slope fields as a way to visualize the general behavior of solutions to a differential equation by plotting the slope of the tangent line at various points in the plane.
Antiderivatives, Differential Equations, and Slope Fields (1).pptLaibaRao4
This document discusses antiderivatives, differential equations, and slope fields. It defines antiderivatives as the inverse operation of differentiation, and shows examples of finding antiderivatives. It explains that solving a differential equation involves isolating the differential and taking the antiderivative to find the indefinite solution. The document also defines slope fields as a way to visualize the general behavior of a differential equation's solutions by plotting the slope of the tangent line at different points in the plane.
This document provides information about solving absolute value equations and inequalities, as well as quadratic equations. It discusses:
1) To solve absolute value equations, you must divide the equation into two equations by treating the expression inside the absolute value bars as both positive and negative.
2) For inequalities, the direction of the inequality sign must be reversed when multiplying or dividing both sides by a negative number.
3) Quadratic equations can be solved by factoring if possible, or using the quadratic formula. The discriminant determines the number of real roots.
I am Harvey L. I am a Computation and System Biology Assignment Expert at nursingassignmenthelp.com. I hold a Ph.D. in Biology, from Bond University, Australia. I have been helping students with their assignments for the past 14 years. I solve assignments related to Computation and System Biology.
Visit nursingassignmenthelp.com or email info@nursingassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Computation and System Biology Assignments.
I am Arcady N. I am a Single Variable Calculus Assignment Expert at mathsassignmenthelp.com. I hold a Master's in Mathematics from, Queen’s University. I have been helping students with their assignments for the past 12 years. I solve assignments related to Single Variable Calculus.
Visit mathsassignmenthelp.com or email info@mathsassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Single Variable Calculus Assignment.
Similar to Chapter 4 EIgen values and eigen vectors.pptx (20)
How to Build a Module in Odoo 17 Using the Scaffold MethodCeline George
Odoo provides an option for creating a module by using a single line command. By using this command the user can make a whole structure of a module. It is very easy for a beginner to make a module. There is no need to make each file manually. This slide will show how to create a module using the scaffold method.
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Tags: Information Security, ISO/IEC 27001, ISO/IEC 42001, Artificial Intelligence, GDPR
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5. More Math
Page 5
not
Thus we want a solutions to 𝐴 − 𝜆𝐼 𝑥 = 0 other than 𝑥 = 0
Recall:
Theorem: any invertible matrix − 𝐴 − 𝜆𝐼 − only has one solution.
Therefore, 𝐴 − 𝜆𝐼 needs to not be invertible.
noninvertible matrices all have a determinant of 0.
Thus det 𝐴 − 𝜆𝐼 needs to equal 0.
Now we move forward by finding 𝜆 such that det | 𝐴 − 𝜆𝐼 | = 0
and
6. What are eigenvalues?
• Given a matrix, A, x is the eigenvector and is the
corresponding eigenvalue if Ax = x
• A must be square and the determinant of A - I must be
equal to zero
Ax - x = 0 iff (A - I) x = 0
• Trivial solution is if x = 0
• The nontrivial solution occurs when det(A - I) = 0
• Are eigenvectors unique?
• If x is an eigenvector, then x is also an eigenvector and is
an eigenvalue
A(x) = (Ax) = (x) = (x)
9. Calculating the Eigenvectors/values
• Expand the det(A - I) = 0 for a 2 x 2 matrix
• For a 2 x 2 matrix, this is a simple quadratic equation with two solutions
(maybe complex)
• This “characteristic equation” can be used to solve for x
0
0
0
det
0
1
0
0
1
det
det
21
12
22
11
22
11
2
21
12
22
11
22
21
12
11
22
21
12
11
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
I
A
21
12
22
11
2
22
11
22
11
4 a
a
a
a
a
a
a
a
10. Eigenvalue example
• Consider,
• The corresponding eigenvectors can be computed as
• For = 0, one possible solution is x = (2, -1)
• For = 5, one possible solution is x = (1, 2)
5
,
0
)
4
1
(
0
2
2
4
1
)
4
1
(
0
4
2
2
1
2
2
21
12
22
11
22
11
2
a
a
a
a
a
a
A
0
0
1
2
2
4
1
2
2
4
0
5
0
0
5
4
2
2
1
5
0
0
4
2
2
1
4
2
2
1
0
0
0
0
0
4
2
2
1
0
y
x
y
x
y
x
y
x
y
x
y
x
y
x
y
x
21. Page 21
The Eigenvectors and Eigenvalues of A and A-1
If 𝜆 is an eigenvalue of A, then 1/𝜆 is an eigenvalue of A-1.
The corresponding eigenvectors are the same.
22. Page 22
The Eigenvectors and Eigenvalues of A and AT
The Eigenvalues of A and AT are the same, but eigenvectors are different.
26. Observe that a 2x2 matrix may have
only one eigenvalue.
Eigenvalues can also be complex numbers.
These facts should not be surprising,
since eigenvalues are solutions of
polynomials.
27. Page 27
Finding Eigenvalues in Real Applications
The eigenvalues of a matrix A can be determined by finding the roots of the characteristic polynomial. Explicit
algebraic formulas for the roots of a polynomial exist only if the degree n is 4 or less. Therefore, for matrices of
order 5 or more, the eigenvalues and eigenvectors cannot be obtained by an explicit algebraic formula and must
be computed by approximate numerical methods.
We would expect that for larger matrices a computer would be used to factor the characteristic polynomials. In
reality, this method is unreliable as roundoff errors cause unpredictable results. Furthermore, before computing
the characteristic polynomial, we need to compute the determinant, also a very expensive process. Efficient,
accurate methods to compute eigenvalues and eigenvectors of arbitrary matrices were not known until the
advent of the QR algorithm in 1961.
Question: how then are eigenvalues found?
Answer: Via iterative processes that transform matrix A into a matrix that is almost an upper triangular
matrix. The more iterations, the better approximation.
Once the value of an eigenvalue is known, the corresponding eigenvectors can be found by finding non-zero
solutions of the eigenvalue equation, that becomes a system of linear equations with known coefficients.
28. Page 28
Example Compute the eigenvalues and eigenvectors for matrix A. How about A2 , A-1 , and A + 4I ?
𝐴 =
2 −1
−1 2