BASIC MATRIX ALGEBRA WITH ALGORITHMS AND APPLICATIONS ROBERT A. LIEBLER CHAPMAN & HALL/CRC A CRC Press Company Boca Raton London New York Washington, D.C.
Contents Preface Examples Major results/proofs xi xiii xv 1 Systems of linear equations and their solution 1 1.1 Recognizing linear systems and solutions 2 Given a system of equations and a collection of variables, determine whether the system is linear in the given variables 2 Given a system of linear equations in n variables, determine whether a given ra-tuple is a solution 3 1.2 Matrices, equivalence and row operations 7 Given a system of linear equations with numerical coefficients, write the associated augmented matrix and vice versa 7 Given a matrix and sequence of elementary row operations, apply the row operations to obtain an equivalent matrix 10 1.3 Echelon forms and Gaussian elimination 17 Given a matrix, decide if it is in reduced row echelon form 17 Given a matrix, apply elementary row operations to obtain an equivalent matrix in reduced row echelon form 18 Solve several linear systems with the same coefficient matrix at once.... 22 1.4 Free variables and general solutions 25 Given an RREF augmented matrix, write the general solution of the associated linear system in terms of the variables from nonpivotal columns of the coefficient matrix, or state that there is no solution. 25 Given a system of linear equations, write an associated augmented matrix, apply elementary row operations to obtain an equivalent RREF matrix and write the general solution* 27 1.5 The vector form of the general solution 30 Given a matrix A and a vector x compute Ax if it exists. Translate between a linear system and the associated matrix equation 30 Compute expressions and verify identities using matrix-vector multiplication and linear combinations 31 Given Ax. = b, with A in an echelon form, write the general solution as a linear combination of basic solutions and the distinguished solution 34 vn
viii Contents. 1.6 Geometric vectors and linear functions 38 Interpret vector arithmetic geometrically. Distances and angles in R 2, R 3.. 38 Transform plots of linear functions and linear systems geometrically.... 41 1.7 Polynomial interpolation 49 Determine the general form of a polynomial function having degree at most n and having m specified functional values, w# < 3 49 2 Matrix number systems 53 2.1 Complex numbers 54 Compute sums, differences, products and quotients in C. Compute the zeros of quadratic polynomials 54 Interpret complex arithmetic in the geometric representation using both the ( rectangular and polar forms 56 2.2 Matrix multiplication 62 Given matrices, compute sums, scalar multiples and products as indicated, or decide the indicated expression does not exist 62 Given an adjacency matrix, sketch the associated labeled digraph and vice versa. Use matrix multiplication to compute the number of walks of specified type 63 Simplify and evaluate matrix expressions using properties of matrix arithmetic 64 Verify complex identities in the matrix representation of C 66 2.3 Auxiliary matrices and matrix inverses 69 Given a polynomial f(x) a square matrix A and a vector v, compute /(A) v with an auxiliary matrix 69 Given an ra-by-n matrix A, compute A~ l or decide that A isn't invertible.. 72 2.4 Symmetric projectors, resolving vectors 77 Given an n-by-r matrix A of rank r, compute PA and the complementary projector P A ±. Explain their geometric meaning 77 Given vectors a, w G R 3, resolve w with respect to a 79 2.5 Least squares approximation 82 Given a set of up to five data points and m < 4, find the (m l)-th degree least squares approximation f(x) to the data 82 2.6 Changing plane coordinates 87 Given two plane coordinate systems specified by their origins and basic unit vectors, compute the coordinates of a point specified in one coordinate system in terms of the other. 87 2.7 The fast Fourier transform and the Euclidean algorithm 95 Given a polynomial g(x) and an integer n > deg(g), write the matrix R g of the "remainder when divided by g(x)." 95 Given a row partitioned Vandermonde matrix, factor it as a block diagonal matrix of smaller Vandermonde matrices and of stacked remainder matrices. r. 98 Given two polynomials, f(x),g(x) compute polynomials a(x),b(x) such that a(x)f(x) + b(x)g(x) equals their greatest common divisor.. 102
Contents. ix Diagonalizable matrices 109 3.1 Eigenvectors and eigenvalues 110 Given a square matrix A and vector v, decide if v is an eigenvector of A.. 110 Given a square matrix A and a number A, decide if A is an eigenvalue of A, and if so, find the associated basic eigenvectors 112 3.2 The minimal polynomial algorithm 117 Given a square matrix A, a vector v and a polynomial f(x), decide if f(x) annihilates the pair A, v. If not, decide the A annihilator of v... 117 Compute the minimal polynomial of a square matrix 121 3.3 Linear recurrence relations 128 Given a digraph T> and a specified walk type, let w n be the number of walks of length n in V and of the specified type. Compute the initial values and a recurrence relation satisfied by the sequence w n... 128 Let b n be the number of binary sequences of length n having no sublists in a given list L and starting and ending in specified ways. Determine the initial values of b n and a recurrence relation satisfied by {6 n } using a transfer matrix 132 3.4 Properties of the minimal polynomial 136 Given an n-by-n matrix (n < 3), write AP = PD, where D is diagonal and P has columns the standard list of basic eigenvectors of A... 136 Given a diagonalizable matrix A and its minimal polynomial, write A as a sum of its components 141 3.5 The sequence {A k } 147 Given an n-by-n matrix A, n < 3, decide if it is diagonalizable and if so, find its spectral radius and a formula for an arbitrary power of A.. 147 Given an n-by-n matrix A, n < 3, describe the evolution of {A k } 148 Given a 3-by-3 numerical matrix with three distinct real eigenvalues, compute the dominant eigenvalue by iteration and give the deflated matrix 152 3.6 Discrete dynamical systems 158 Given a three-state linear discrete dynamical system, write its transition matrix A. Compute the spectral radius of A and determine the system's long-term behavior. Identify any stable steady states 158 3.7 Matrix compression with components 167 Given r < 2 and a 4 by 4 matrix with distinct eigenvalues, compute its components and write an optimal rank r approximation to A... 167 Given data involving three attributes, standardize it and compute the associated correlation matrix. Determine the first principal component and determine what percentage of the data variance it explains... 169
x Contents. 4 Determinants 177 4.1 Area and composition of linear functions 178 Given square matrix A, factor it as a product of elementary matrices and RREF(A). Use a sequence of transform plots to realize the associated linear function /A as a composition of elementary linear functions 178 Given a transform plot /A : R 2 > R 2 that tracks a figure, use area and orientation to compute det(a) 180 4.2 Computing determinants 185 Compute the determinant of an n-by-n(n < 4) matrix using the cofactor definition 185 Compute the determinant of an n-by-n(n < 4) matrix using Gaussian elimination and the block triangular formula 189 4.3 Fundamental properties of determinants 196 Compute the determinant of an n-by-n (n < 5) matrix using arbitrary cofactor expansions, matrix reduction or block matrices as required.. 196 Use properties of determinants to verify formulas without computation... 199 Given a 2-by-2 integer matrix, compute its Smith normal form and interpret the associated lattice graphically 202 4.4 Further applications 207 Find the characteristic polynomial/equation of an n-by-n matrix (n < 4).. 207 Use Cramer's rule to solve the linear system Ax = b when appropriate... 209 A pedagogical postscript 213 Appendix: The abstract setting 217 Selected practice problem answers 231 Index 239