Chapter 4 Linear Algebra Problems

Chapter 4 Linear Algebra Problems
Solving Applied Mathematical Problems with MATLAB CRC/Taylor & Francis Press Chinese version by Tsinghua University Press PPT by Wenbin Dong and Jun Peng, Northeastern University, PRC Proofread by Dingyu Xue & YangQuan Chen 星期四, , 12:40:54 Slide 1 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Chapter 4 Linear Algebra Problems
Inputting Special Matrices Fundamental Matrix Operations Fundamental Matrix Transformations Solving Matrix Equations Nonlinear Functions and Matrix Function Evaluations Chapter summary 星期四, , 12:40:54 Slide 2 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

4.1 Inputting Special Matrices
Numerical matrix input Defining symbolic matrices 星期四, , 12:40:54 Slide 3 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

4.1.1 Numerical matrix input
Matrices of zeros, ones and identity matrices Matrices with random elements Diagonal matrices Hankel matrices Hilbert matrices and their inverses Vandermonde matrices Companion matrices 星期四, , 12:40:54 Slide 4 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Matrices of zeros, ones and identity matrices
The syntax of matrices of zero matrix, identity matrix and matrix of ones square matrix 星期四, , 12:40:54 Slide 5 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Multi-dimensional arrays and other data forms are also supported
rectangular matrix with the same size of B Multi-dimensional arrays and other data forms are also supported 星期四, , 12:40:54 Slide 6 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Example 4.1 Generate a zero matrix A, and generate a expanded unit matrix B with the same dimension as matrix A. MATLAB solutions: Note: zeros() and ones() can also be used to define multi-dimensional arrays. 星期四, , 12:40:54 Slide 7 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Matrices with random elements
Each element of matrix satisfies a uniform distribution in The syntax: An uniformly distributed random matrix An random 星期四, , 12:40:54 Slide 8 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Declare a random matrix of size
Function rand() can also be used to define multi-dimensional random arrays Declare a random matrix of size The uniform distribution matrix over [a,b] interval can be generated with 星期四, , 12:40:54 Slide 9 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Define a standard normal distributed N(0,1) random matrices
Generating N(m,s2) random numbers 星期四, , 12:40:54 Slide 10 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Diagonal matrices Mathematical description to a diagonal matrix
where all the non-diagonal elements are 0 星期四, , 12:40:54 Slide 11 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Define a matrix from given vector
Extract diagonal vector from a given matrix Define the kth diagonal Note: k can be assigned to negative integers 星期四, , 12:40:54 Slide 12 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Example 4.2 Different calling formats of diag() function
Generate a diagonal matrix: Diagonal elements extraction: Input the corresponding diagonal line V 星期四, , 12:40:54 Slide 13 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Generate the tri-diagonal matrix:
The matrix can be constructed with 星期四, , 12:40:54 Slide 14 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Construct a block diagonal matrix:
The syntax 星期四, , 12:40:54 Slide 15 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Hankel matrices The general form of the Hankel matrix
Given two vectors and 星期四, , 12:40:54 Slide 16 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

where vector is the first column, vector is the last row.
Construct an upper-triangular Hankel matrix 星期四, , 12:40:54 Slide 17 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Example 4.3 Establish the following Hankel matrices MATLAB solutions:
星期四, , 12:40:54 Slide 18 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Hilbert matrices and their inverses
The general form of the Hilbert matrix where 星期四, , 12:40:54 Slide 19 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

The syntax for generating a Hilbert matrix
The syntax for generating inverse Hilbert matrix Note: Hilbert matrices are very close to singular matrices, symbolic computation is recommended! 星期四, , 12:40:54 Slide 20 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Vandermonde matrices Mathematical description to the Vandermonde matrix where Generating a Vandermonde matrix 星期四, , 12:40:54 Slide 21 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Example 4.4 Establish a Vandermonde matrix MATLAB solutions:

Companion matrices The companion matrix of a monic polynomial
Math form 星期四, , 12:40:54 Slide 23 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

The syntax for generating a companion matrix
where is a polynomial coefficient vector 星期四, , 12:40:54 Slide 24 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Example 4.5 Consider a polynomial Find its companion matrix
MATLAB solutions: 星期四, , 12:40:54 Slide 25 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

4.1.2 Defining symbolic matrices
Convert a numerical matrix A into a symbolic one, for analytical solutions The syntax for converting to a symbolic matrix Overload matrix entering compan, hankel, vander 星期四, , 12:40:54 Slide 26 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Construct a symbolic companion matrix:

Example 4.6 Given establish its companion matrix MATLAB solutions:

Construct a symbolic Hankel matrix:

Construct a symbolic Vandermonde matrix:
All the three functions should be saved Avoid conflict with existing ones 星期四, , 12:40:54 Slide 30 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

4.2 Fundamental Matrix Operations
Basic concepts and properties of matrices Matrix inversion and generalized matrix inversion Matrix eigenvalue problems 星期四, , 12:40:54 Slide 31 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

4.2.1 Basic concepts and properties of matrices
Determinant Trace Rank Norms Characteristic polynomials Evaluation of polynomial matrices Conversion between symbolic and numerical polynomials 星期四, , 12:40:54 Slide 32 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Determinant The determinant of matrix is defined as The syntax
Note: this function applies both to symbolic and numerical matrices . 星期四, , 12:40:54 Slide 33 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Example 4.7 Compute the determinant of a given matrix

Example 4.8 Calculate analytically the determinant of a Hilbert matrix
MATLAB solutions: Result: 星期四, , 12:40:54 Slide 35 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Trace The trace of a square matrix is defined as The syntax

Rank The rank of matrix A is defined as
is the column rank and is the row rank. The syntax Symbolic or numerical Numerical rank with error tolerance of 星期四, , 12:40:54 Slide 37 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Example 4.9 Find the rank of MATLAB solutions:

Example 4.10 Determine the rank of a Hilbert matrix with numerical or analytical methods . MATLAB solutions: Numerical method: Analytical method: Numerical method may give misleading results 星期四, , 12:40:54 Slide 39 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Scalar norms For a scalar function to be a vector norm of vector , the following 3 conditions must be satisfied: and if and only if a is any given scalar For any vectors and , the following holds 星期四, , 12:40:54 Slide 40 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

This p-norm satisfies all the three conditions:
and is the notation of a vector norm. 星期四, , 12:40:54 Slide 41 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Matrix norms For any non-zero vector , the norm is
Commonly used matrix norms: 星期四, , 12:40:54 Slide 42 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Note: norm() applies only to numerical matrices
where is the eigenvalues of the matrix , while is the maximum eigenvalue of The syntax Default for options could be 1,2, inf, ‘fro’ Note: norm() applies only to numerical matrices 星期四, , 12:40:54 Slide 43 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Compute various norms of

Characteristic polynomials
The characteristic polynomial of a matrix is The polynomial is referred to as the characteristic polynomial of matrix The syntax 星期四, , 12:40:54 Slide 45 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Example 4.11 Find the characteristic polynomial of matrix
Direct computation: 星期四, , 12:40:54 Slide 46 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Using Symbolic Math Toolbox:
Leverrier-Faddeev recursive algorithm: where Loop structure can be used 星期四, , 12:40:54 Slide 47 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Main body of the function
A new MATLAB function Main body of the function 星期四, , 12:40:54 Slide 48 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

code continued from the last slide

Example 4.12 Given vector , establish a Hankel matrix and find its characteristic polynomial. MATLAB solutions: 星期四, , 12:40:54 Slide 50 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

If the polynomial can be expressed as
Then and Loops can be used 星期四, , 12:40:54 Slide 51 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Construct an MATLAB function

Example 4.13 Extract the coefficients of the characteristic polynomial of matrix MATLAB solutions: 星期四, , 12:40:54 Slide 53 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Evaluation of polynomial matrices
Mathematical form of polynomial matrices : The syntax: where is the coefficients in descending order of s of the polynomial. 星期四, , 12:40:54 Slide 54 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Definition of dot operation: The syntax
Get a polynomial in p (Symbolic Math): 星期四, , 12:40:54 Slide 55 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Cayley-Hamilton Theorem:
If the characteristic polynomial of a matrix is given by Then, , that is 星期四, , 12:40:54 Slide 56 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Example 4.14 Verify the Cayley-Hamilton theorem using a Vandermonde matrix as an example. MATLAB solutions (erroneous): Using poly1() function: 星期四, , 12:40:54 Slide 57 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Conversion between symbolic and numerical polynomials
Given a vector Convert a numerical polynomial to symbolic expression: Convert a symbolic expression to a vector: 星期四, , 12:40:54 Slide 58 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Example 4.15 Using both numerical and symbolic forms to express
Convert to the symbolic polynomial: Convert to the numerical form of a polynomial: 星期四, , 12:40:54 Slide 59 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

4.2.2 Matrix inversion and generalized inverse of a matrix
Inverse matrix Generalized matrix inverse 星期四, , 12:40:54 Slide 60 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Inverse matrix Mathematical description to inverse matrix
where is an non-singular square matrix, then, The syntax 星期四, , 12:40:54 Slide 61 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Example 4.16 Inverse for the given Hilbert matrix Hilbert matrix
Use invhilb() function to directly get the inverse matrix: 星期四, , 12:40:54 Slide 62 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Hilbert matrix: 星期四, 2008-4- 24, 12:40:54

Hilbert inverse matrix:
For symbolic matrix: Hilbert inverse matrix: 星期四, , 12:40:54 Slide 64 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Example 4.17 Inverse matrix for singular matrix MATLAB solutions:
Using Symbolic Math Toolbox: 星期四, , 12:40:54 Slide 65 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Example 4.18 Derive the inverse matrix of the Hankel matrix

Generalized matrix inverse
For singular and/or rectangular matrix If , then is called the generalized inverse of , denoted by Not unique Define the norm minimization index as 星期四, , 12:40:54 Slide 67 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Moore-Penrose generalized inverse
Conditions for a matrix to be a Moore-Penrose generalized inverse of matrix : (i) (ii) (iii) and are Hermitian symmetrical Denote Unique and meaningful 星期四, , 12:40:54 Slide 68 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

The syntax: Evaluate the Moore-Penrose pseudo-inverse
Evaluate the inverse numerically with precision of 星期四, , 12:40:54 Slide 69 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Example 4.19 Use pinv() to compute the inverse of MATLAB solutions:

Verify the 3 conditions for Moore-Penrose generalized inverse:

Example 4.20 Find the rank and Moore-Penrose inverse for Rank:

Generalized inverse and verification:

4.2.3 Matrix eigenvalue problems
Eigenvalues and eigenvectors of a matrix Generalized eigenvalues and eigenvectors 星期四, , 12:40:54 Slide 74 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Eigenvalues and eigenvectors
Mathematical description: Non-zero vector is an eigenvector, scalar is an eigenvalue. The syntax or 星期四, , 12:40:54 Slide 75 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Example 4.21 Compute the eigenvalues and eigenvectors for
Direct computation: 星期四, , 12:40:54 Slide 76 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Find eigenvalues and eigenvectors
Precise solution: Find eigenvalues and eigenvectors Direct solution: Analytical/precise solution: 星期四, , 12:40:54 Slide 77 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Generalized eigenvalues and eigenvectors
Mathematical description: Non-zero vector is an eigenvector, scalar is an eigenvalue is a symmetrical positive-definite matrix The syntax or 星期四, , 12:40:54 Slide 78 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Example 4.22 Solve generalized eigenvalues and generalized eigenvectors MATLAB solutions: 星期四, , 12:40:54 Slide 79 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

4.3 Fundamental Matrix Transformations
Similarity transformations and orthogonal matrices Triangular and Cholesky decompositions Jordan transformations Singular value decompositions 星期四, , 12:40:54 Slide 80 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

4.3.1 Similarity transformations and orthogonal matrices
Mathematical description to similarity transformation Mathematical description to orthogonal matrix The syntax 星期四, , 12:40:54 Slide 81 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Example 4.23 Compute the orthogonal basis for matrix
MATLAB solutions and verification: 星期四, , 12:40:54 Slide 82 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Example 4.24 Given get the orthogonal basis matrix and verify the its properties. 星期四, , 12:40:54 Slide 83 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

If there exists non-singular matrix
and column vector Matrix can be transformed into a companion-like matrix 星期四, , 12:40:54 Slide 84 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Example 4.25 Given transform the matrix into a companion matrix.

continued from the previous slide
The standard companion form: 星期四, , 12:40:54 Slide 86 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

4.3.2 Triangular and Cholesky decompositions
Triangular decompositions Cholesky decomposition of symmetrical matrices Positive-definiteness, regular matrix: definitions and tests 星期四, , 12:40:54 Slide 87 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Triangular decompositions
Mathematical description where 星期四, , 12:40:54 Slide 88 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Recursive computation formula:
Initial values for the recursion: 星期四, , 12:40:54 Slide 89 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

The MATLAB function syntax: LU decomposition
is the permutation matrix, 星期四, , 12:40:54 Slide 90 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Symbolic MATLAB function, @sym

Example 4.26 Given Two methods to call lu() function MATLAB solutions:
step 1: 星期四, , 12:40:54 Slide 92 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Verification: Symbolic solution step 2: 星期四, 2008-4- 24, 12:40:54

Cholesky decomposition of symmetrical matrices
Mathematical description where is a symmetrical matrix 星期四, , 12:40:54 Slide 94 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Cholesky decomposition algorithm for symmetrical matrix:

Overloaded function chol() , @sym
Note: Cholesky decomposition cannot be performed for asymmetrical matrices. 星期四, , 12:40:54 Slide 96 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Example 4.27 Cholesky decomposition for MATLAB solution:

Positive-definiteness, regular matrix: definitions and tests
Mathematical description Positive-definite matrix: all the determinants of the main sub-matrices have the same sign 星期四, , 12:40:54 Slide 98 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

The syntax for checking positive-definiteness
Regular matrix where is a complex square matrix, is the Hermite transpose of . The syntax for checking regular matrix 星期四, , 12:40:54 Slide 99 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Example 4.28 Given check if it is a positive definite matrix, and obtain its Cholesky decomposition matrix. MATLAB solutions: 星期四, , 12:40:54 Slide 100 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

4.3.3 Jordan transformations
Jordanian decomposition can deal with the matrices containing repeated eigenvalues. The syntax: only the Jordanian returned Jordanian and generalized vector matrix 星期四, , 12:40:54 Slide 101 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Example 4.29 Given determine the eigenvalues and eigenvectors of matrix with numerical and analytical methods 星期四, , 12:40:54 Slide 102 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Numerical method: Analytical method: 星期四, 2008-4- 24, 12:40:54

Example 4.30 Perform Jordan decomposition for matrix MATLAB solutions:

Example 4.31 Perform the Jordan decomposition of matrix

Real matrices can be given
Result 星期四, , 12:40:54 Slide 106 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

4.3.4 Singular value decompositions
Mathematical description In theory Definition of singular values: where are nonnegative eigenvalues. 星期四, , 12:40:54 Slide 107 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Example 4.32 Given compute the rank , where Calculate
Direct MATLAB solutions: 星期四, , 12:40:54 Slide 108 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Mathematical description of singular value decomposition (SVD)
where is an matrix, and are orthogonal matrices. is a diagonal matrix. The diagonal elements satisfy 星期四, , 12:40:54 Slide 109 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

The syntax only singular value decomposition required

The condition number of the matrix
where and are the maximum singular value, and are the minimum singular value. The syntax: 星期四, , 12:40:54 Slide 111 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Example 4.33 SVD for matrix MATLAB solutions: Condition number:

Example 4.34 SVD for , and then verify the results. MATLAB solutions:

4.4 Solving Matrix Equations
Solutions to linear algebraic equations Solutions to Lyapunov equations Solutions to Sylvester equations Solutions to Riccati equations 星期四, , 12:40:54 Slide 114 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

4.4.1 Solutions to linear algebraic equations
Mathematical description Given matrices 星期四, , 12:40:54 Slide 115 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Unique solutions Existence theorem:
When and , a unique solution exists The syntax NOTE: the symbolic method is recommended 星期四, , 12:40:54 Slide 116 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Example 4.35 Solve the linear algebraic equations MATLAB solutions

Equations with infinite number of solutions
The checking matrix When , there are infinite number of solutions. 星期四, , 12:40:54 Slide 118 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

where are arbitrary constants The syntax
General solutions where are arbitrary constants The syntax Get matrix's null matrix Solve matrix's null matrix in regular form 星期四, , 12:40:54 Slide 119 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Example 4.36 Find the solutions to the equations MATLAB solutions:
Check solvability: 星期四, , 12:40:54 Slide 120 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

The general solution Result: 星期四, 2008-4- 24, 12:40:54

Equations with no solutions
If , use only Moore-Penrose generalized inverse to get a solution in the least squares sense as which minimizes the error norm 星期四, , 12:40:54 Slide 122 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Example 4.37 Solve MATLAB solutions: Least squares solution:

4.4.2 Solutions to Lyapunov equations
Continuous Lyapunov equations Analytical solutions to Lyapunov equations Discrete Lyapunov equations Sylvester (generalized Lyapunov) equation 星期四, , 12:40:54 Slide 124 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Continuous Lyapunov equations
Mathematical description Temporarily assume that is a symmetrical positive definite matrix The syntax, Control Systems Toolbox: 星期四, , 12:40:54 Slide 125 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Example 4.38 Given solve Lyapunov equation and verify results

Analytical solutions to Lyapunov equations
Mathematical description the matrices are rearranged into vectors such that 星期四, , 12:40:54 Slide 127 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

The Lyapunov equation can be rewritten as: 星期四, , 12:40:54 Slide 128 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

denotes the Kronecker product of matrices and

Example 4.39 Given obtain the analytical solution of Lyapunov equation. MATLAB solutions: 星期四, , 12:40:54 Slide 130 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Example 4.40 Assume where is not a symmetrical real matrix,
Find whether there are solutions to this Lyapunov equation. 星期四, , 12:40:54 Slide 131 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

MATLAB solutions: Conclusion: matrix can be any matrix if the generalized energy concept in Lyaponov stability analysis is not considered. 星期四, , 12:40:54 Slide 132 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Discrete Lyapunov equations
Mathematical description: The syntax: 星期四, , 12:40:54 Slide 133 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Example 4.41 Solve the discrete Lyapunov equation and verify the result. MATLAB solutions: 星期四, , 12:40:54 Slide 134 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

4.4.3 Solutions to Sylvester equations
Mathematical description of generalized Lyapunov equation The syntax The analytical solutions using Kronecker products: 星期四, , 12:40:54 Slide 135 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Construct an MATLAB function

The syntax Continuous Lyapunov equation
Discrete Lyapunov equation, rewritten as Sylvester equation 星期四, , 12:40:54 Slide 137 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Example 4.42 Solve the following Sylvester equation
Numerical solution: 星期四, , 12:40:54 Slide 138 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Analytical solution: 星期四, 2008-4- 24, 12:40:54

Example 4.43 Obtain the analytical solution for MATLAB solutions:

Example 4.44 Solve the Sylvester equation with MATLAB solutions:

4.4.4 Solutions to Riccati equations
Mathematical description of Riccati equations The syntax NOTE: function are( ) is numerical based. ARE stands for “algebraic Riccati equation.” 星期四, , 12:40:54 Slide 142 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Example 4.45 Solve the following Riccati equation with:

MATLAB commands: Result:
Remaining problem: only one solution found for quadratic equation, see Chapter 6 星期四, , 12:40:54 Slide 144 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

4.5 Nonlinear Functions and Matrix Function Evaluations
Element-by-element computations Matrix function evaluations 星期四, , 12:40:54 Slide 145 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

4.5.1 Element-by-element computations
Commonly used element-by-element nonlinear functions 星期四, , 12:40:54 Slide 146 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Continued from the previous page

Example 4.46 Given find the element-by-element matrix exponential and sinusoidal functions MATLAB commands: 星期四, , 12:40:54 Slide 148 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

4.5.2 Matrix function evaluations
Computations of matrix exponentials Trigonometric functions of matrices General matrix functions 星期四, , 12:40:54 Slide 149 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Computations of matrix exponentials
19 numerical solution methods to compute Moler C B, Van Loan C F. Nineteen dubious ways to compute the exponential of a matrix. SIAM Review, 1979, 20: The built-in MATLAB function syntax: Also works for symbolic computation 星期四, , 12:40:54 Slide 150 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Example 4.47 Given solve , and analytically Numerical solution:

The expm() function is also applicable to symbolic matrices
Result: 星期四, , 12:40:54 Slide 152 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Example 4.48 Given calculate MATLAB solutions

Jordanian transformation technique
Then, the exponential function of the original matrix becomes 星期四, , 12:40:54 Slide 154 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Trigonometric functions of matrices
There is no MATLAB functions for trigonometric matrix operations The syntax of matrix function for any nonlinear matrix functions 星期四, , 12:40:54 Slide 155 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Example 4.49 Given solve . MATLAB solutions: 星期四, 2008-4- 24, 12:40:54

Taylor series expansion of sine function:
Construct an MATLAB function for sinusoidal matrix 星期四, , 12:40:54 Slide 157 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Example 4.50 Given solve MATLAB solution: 星期四, 2008-4- 24, 12:40:54

Alternative trigonometric matrix
From Euler formula: We can derive: Applies for matrices 星期四, , 12:40:54 Slide 159 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Example 4.51 Given solve MATLAB solution: 星期四, 2008-4- 24, 12:40:54

Example 4.52 Given with repeating eigenvalues, Solve and .

MATLAB solutions: 星期四, 2008-4- 24, 12:40:54

General matrix functions
Jordan block where is an eigenvalue with mutiplicity is a nilpotent matrix, that is when Matrix function : 星期四, , 12:40:54 Slide 163 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Decompose any known matrix with Jordan matrices
The arbitrary function of this matrix is: 星期四, , 12:40:54 Slide 164 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

MATLAB function used to find analytical solution of any function of any matrix

where is a symbolic variable, is the function of variable .
The syntax where is a symbolic variable, is the function of variable . 星期四, , 12:40:54 Slide 168 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Example 4.53 Given compute the matrix function

Complicated results, upper-left corner term
MATLAB solutions: Complicated results, upper-left corner term 星期四, , 12:40:54 Slide 170 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

When the precise numerical matrix
Collecting terms: Simplified result When the precise numerical matrix 星期四, , 12:40:54 Slide 171 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Chapter Summary List of Relevant MATLAB Functions

We can use MATLAB to get numerical and/or symbolic/analytical solutions, e.g., for matrix determinant, trace, rank, norms, characteristic polynomial, inverse, generalized inverse, eigenvalue, eigenvector etc. We also learned several matrix decompositions such as LU decomposition, orthogonal decomposition, Cholesky decomposition , Jordan decomposition, singular value decomposition etc. We introduced how to use MATLAB commands to directly obtain these decompositions. 星期四, , 12:40:54 Slide 176 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

This chapter also introduced the solvability conditions of linear simultaneous equations and how to handle the cases of the unique solution, infinite number of solutions and no solution. We learned how to use MATLAB to construct the general solution family when there are infinite number of solutions. Least squares based solution is also introduced where there exists no solution. Solution methods for continuous and discrete Lyapunov equation, Sylvester equation including numerical method and Kronecker product based analytical method. Algebraic Riccati equation is also introduced. 星期四, , 12:40:54 Slide 177 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

This chapter introduced concepts of point-wise function evaluation and matrix function evaluation. For matrix exponential function, matrix trigonometry functions etc., both numerical and analytical methods are given. For general form of matrix functions, such as a Jordan decomposition based numerical method is presented with a ready MATLAB function developed and demonstrated. 星期四, , 12:40:54 Slide 178 (of 178) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Chapter 4 Linear Algebra Problems

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