Matrix Analysis For Scientists And Engineers

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Format: Paperback
Pub. Date: 2004-12-29
Publisher(s): SIAM
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Summary

Matrix Analysis for Scientists and Engineers provides a blend of undergraduate- and graduate-level topics in matrix theory and linear algebra that relieves instructors of the burden of reviewing such material in subsequent courses that depend heavily on the language of matrices. Consequently, the text provides an often-needed bridge between undergraduate-level matrix theory and linear algebra and the level of matrix analysis required for graduate-level study and research. The text is sufficiently compact that the material can be taught comfortably in a one-quarter or one-semester course. Throughout the book, the author emphasizes the concept of matrix factorization to provide a foundation for a later course in numerical linear algebra. The author addresses connections to differential and difference equations as well as to linear system theory and encourages instructors to augment these examples with other applications of their own choosing.

Author Biography

Alan J. Laub is a Professor in the Departments of Computer Science and Applied Science at the University of California, Davis.

Table of Contents

Preface xi
Introduction and Review
1(6)
Some Notation and Terminology
1(2)
Matrix Arithmetic
3(1)
Inner Products and Orthogonality
4(1)
Determinants
4(3)
Vector Spaces
7(10)
Definitions and Examples
7(2)
Subspaces
9(1)
Linear Independence
10(3)
Sums and Intersections of Subspaces
13(4)
Linear Transformations
17(12)
Definition and Examples
17(1)
Matrix Representation of Linear Transformations
18(1)
Composition of Transformations
19(1)
Structure of Linear Transformations
20(2)
Four Fundamental Subspaces
22(7)
Introduction to the Moore--Penrose Pseudoinverse
29(6)
Definitions and Characterizations
29(1)
Examples
30(1)
Properties and Applications
31(4)
Introduction to the Singular Value Decomposition
35(8)
The Fundamental Theorem
35(3)
Some Basic Properties
38(2)
Row and Column Compressions
40(3)
Linear Equations
43(8)
Vector Linear Equations
43(1)
Matrix Linear Equations
44(3)
A More General Matrix Linear Equation
47(1)
Some Useful and Interesting Inverses
47(4)
Projections, Inner Product Spaces, and Norms
51(14)
Projections
51(3)
The four fundamental orthogonal projections
52(2)
Inner Product Spaces
54(3)
Vector Norms
57(2)
Matrix Norms
59(6)
Linear Least Squares Problems
65(10)
The Linear Least Squares Problem
65(2)
Geometric Solution
67(1)
Linear Regression and Other Linear Least Squares Problems
67(3)
Example: Linear regression
67(2)
Other least squares problems
69(1)
Least Squares and Singular Value Decomposition
70(1)
Least Squares and QR Factorization
71(4)
Eigenvalues and Eigenvectors
75(20)
Fundamental Definitions and Properties
75(7)
Jordan Canonical Form
82(3)
Determination of the JCF
85(4)
Theoretical computation
86(2)
On the +1's in JCF blocks
88(1)
Geometric Aspects of the JCF
89(2)
The Matrix Sign Function
91(4)
Canonical Forms
95(14)
Some Basic Canonical Forms
95(4)
Definite Matrices
99(3)
Equivalence Transformations and Congruence
102(2)
Block matrices and definiteness
104(1)
Rational Canonical Form
104(5)
Linear Differential and Difference Equations
109(16)
Differential Equations
109(9)
Properties of the matrix exponential
109(3)
Homogeneous linear differential equations
112(1)
Inhomogeneous linear differential equations
112(1)
Linear matrix differential equations
113(1)
Modal decompositions
114(1)
Computation of the matrix exponential
114(4)
Difference Equations
118(2)
Homogeneous linear difference equations
118(1)
Inhomogeneous linear difference equations
118(1)
Computation of matrix powers
119(1)
Higher-Order Equations
120(5)
Generalized Eigenvalue Problems
125(14)
The Generalized Eigenvalue/Eigenvector Problem
125(2)
Canonical Forms
127(3)
Application to the Computation of System Zeros
130(1)
Symmetric Generalized Eigenvalue Problems
131(2)
Simultaneous Diagonalization
133(2)
Simultaneous diagonalization via SVD
133(2)
Higher-Order Eigenvalue Problems
135(4)
Conversion to first-order form
135(4)
Kronecker Products
139(12)
Definition and Examples
139(1)
Properties of the Kronecker Product
140(4)
Application to Sylvester and Lyapunov Equations
144(7)
Bibliography 151(2)
Index 153

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