ABSTRACT
Linear Algebra 1 Vectors, Matrices, and Systems of Linear Equations Jane Day . . . . . . 1-1
2 Linear Independence, Span, and Bases Mark Mills . . . . . . . . . . . . . . . . . . . . . 2-1
3 Linear Transformations Francesco Barioli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-1
4 Determinants and Eigenvalues Luz M. DeAlba. . . . . . . . . . . . . . . . . . . . . . . . . . . 4-1 5 Inner Product Spaces, Orthogonal Projection, Least Squares,
and Singular Value Decomposition Lixing Han and Michael Neumann 5-1
6 Canonical Forms Leslie Hogben . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-1
7 Other Canonical Forms Roger A. Horn and Vladimir V. Sergeichuk . . . . . 7-1
8 Unitary Similarity, Normal Matrices, and Spectral Theory Helene Shapiro. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-1
9 Hermitian and Positive Definite Matrices Wayne Barrett . . . . . . . . . . . . . 9-1
10 Nonnegative and Stochastic Matrices Uriel G. Rothblum . . . . . . . . . . . . . . 10-1
11 Partitioned Matrices Robert Reams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-1
Topics in Linear Algebra 12 Schur Complements Roger A. Horn and Fuzhen Zhang . . . . . . . . . . . . . . . . . . . 12-1
13 Quadratic, Bilinear, and Sesquilinear Forms Raphael Loewy . . . . . . . . . . 13-1
14 Multilinear Algebra J. A. Dias da Silva and Armando Machado . . . . . . . . . 14-1
15 Tensors and Hypermatrices Lek-Heng Lim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15-1
16 Matrix Equalities and Inequalities Michael Tsatsomeros . . . . . . . . . . . . . . . 16-1
17 Functions of Matrices Nicholas J. Higham . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17-1
18 Matrix Polynomials Jo¨rg Liesen and Christian Mehl . . . . . . . . . . . . . . . . . . . . . . 18-1
19 Matrix Equations Beatrice Meini . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19-1
20 Invariant Subspaces G. W. Stewart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20-1
21 Matrix Perturbation Theory Ren-Cang Li . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-1
22 Special Types of Matrices Albrecht Bo¨ttcher and Ilya Spitkovsky . . . . . . . . 22-1
23 Pseudospectra Mark Embree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23-1
24 Singular Values and Singular Value Inequalities Roy Mathias . . . . . . . . 24-1
25 Numerical Range Chi-Kwong Li. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25-1
26 Matrix Stability and Inertia Daniel Hershkowitz . . . . . . . . . . . . . . . . . . . . . . . . . 26-1
27 Generalized Inverses of Matrices Yimin Wei . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27-1
28 Inverse Eigenvalue Problems Alberto Borobia . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28-1
29 Totally Positive and Totally Nonnegative Matrices Shaun M. Fallat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29-1
30 Linear Preserver Problems Peter Sˇemrl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30-1
31 Matrices over Finite Fields J. D. Botha . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31-1
32 Matrices over Integral Domains Shmuel Friedland . . . . . . . . . . . . . . . . . . . . . . 32-1
33 Similarity of Families of Matrices Shmuel Friedland . . . . . . . . . . . . . . . . . . . . 33-1
34 Representations of Quivers and Mixed Graphs Roger A. Horn and Vladimir V. Sergeichuk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34-1
35 Max-Plus Algebra Marianne Akian, Ravindra Bapat, Ste´phane Gaubert . . 35-1
36 Matrices Leaving a Cone Invariant Bit-Shun Tam and Hans Schneider 36-1
37 Spectral Sets Catalin Badea and Bernhard Beckermann . . . . . . . . . . . . . . . . . . . 37-1
1 Vectors, Matrices, and Systems of Linear Equations Jane Day . . . . . . 1-1
2 Linear Independence, Span, and Bases Mark Mills . . . . . . . . . . . . . . . . . . . . . 2-1
3 Linear Transformations Francesco Barioli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-1
4 Determinants and Eigenvalues Luz M. DeAlba. . . . . . . . . . . . . . . . . . . . . . . . . . . 4-1 5 Inner Product Spaces, Orthogonal Projection, Least Squares,
and Singular Value Decomposition Lixing Han and Michael Neumann 5-1
6 Canonical Forms Leslie Hogben . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-1
7 Other Canonical Forms Roger A. Horn and Vladimir V. Sergeichuk . . . . . 7-1
8 Unitary Similarity, Normal Matrices, and Spectral Theory Helene Shapiro. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-1
9 Hermitian and Positive Definite Matrices Wayne Barrett . . . . . . . . . . . . . 9-1
10 Nonnegative and Stochastic Matrices Uriel G. Rothblum . . . . . . . . . . . . . . 10-1
11 Partitioned Matrices Robert Reams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-1
Throughout this chapter, F will denote a field. The references [Lay12], [Leo10], and [SIF08] are good sources for more detail about much of the material in this chapter. They discuss primarily the field of real numbers, but the proofs are usually valid for any field. Vector spaces over a general field are presented in [FIS03]
Vectors are used in many applications. They often represent quantities that have both direction and magnitude, such as velocity or position relative to the origin. They can appear as functions, as n-tuples of scalars, matrices, or in other disguises. Whenever objects can be added and multiplied by scalars, they may be elements of some vector space. In this section, we formulate a general definition of vector space and establish its basic properties. An element of a field, such as the real numbers or the complex numbers, is called a scalar to distinguish it from a vector.
