PREFACE
WARMUP
Clear Thinking
Logic
Proofs
Sets
Defining Sets by Enumeration
Common Sets
Sets and Properties
Subsets
Boolean Operations
Equality of Sets
Scalars
Sigma Notation
The Geometry of Linear Systems
Using MINIMAT
Assignment Statements
Complex Numbers
Expressions
Scalar Built-in Functions
.M Functions
Format Command
Control Structures
MATRIX OPERATIONS
Matrices Defined
Additive Operations on Matrices
Multiplicative Operations
Inverses
Transpose
Diagonal Matrices
Triangular Matrices
Matrices of Matrices
Using MINIMAT
Creating New Matrices
White Space
Zero Matrices
Random Matrices
Identity Matrix
Confirming Laws
Transpose
Inverses and Powers
Diagonal Matrices
Triangular Matrices
Matrices of Matrices
Submatrices, Colon Notation
Entrywise Operations
Some Computer Exercises
More Exercises
Preview of the Exponential
Geometric Series
INVERTIBLE MATRICES
Elementary Row Operations
Elementary Matrices
Using MINIMAT
Reduced Tow Echelon Form
Gauss-Jordan Elimination
Computing the Multiplier
Using MINIMAT
How to Invert
Using MINIMAT
Elementary Column Operations
Permutation Matrices
Equivalence
Using MINIMAT
SUBSPACES
Linear Systems
Using MINIMAT
Null Space and Range
Set Equality
Using MINIMAT
Subspaces
New Subspaces from Old
Bases
Basis for the Null Space
Basis for the Range
Using MINIMAT
Bases and Biequivalence
Using MINIMAT
The Range from the RREF
Using MINIMAT
Random Solutions
Co-bases(*)
RANK AND DIMENSION
The Definition of Dimension
Using MINIMAT
Existence
Using MINIMAT
The Analogy
Rank and Nullity
Using MINIMAT
One-Sided Inverses
Using MINIMAT
Equivalence
Uniqueness of the RREF
More Exercises
Characterizations of the Rank
A Block Inverse Formula
Geometry and Independence
Matrix Representation on a Subspace
Real Rank vs. Complex Rank
GEOMETRY
Inner Products and Norms
Real Inner Products
Complex Inner Products
Norms
Geometric Interpretation
Unitary Matrices
Orthonormal Bases
The Gram-Schmidt Decomposition
Positive Triangular Matrices
The Gram-Schmidt Process
Geometric Interpretation
Using MINIMAT
Projection (General)
Projection (Orthogonal)
Using MINIMAT
Least Squares
The Best Approximate Solution
The Closest Point
Using MINIMAT
More Exercises
Submultiplicative Inequality
Norms
Pauli Matrices and Quaternions
DETERMINANTS-I
Permutations
Sign of a Permutation
Transpositions
Using MINIMAT
Determinant Defined
Easy Properties
Computing Determinants
Using MINIMAT
More Exercises
Wedge Product
Real Equivalence
DIAGONALIZATION
Similarity
Eigenvalues and Eigenvectors
Computing Eigenvalues
Using MINIMAT
The Characteristic Polynomial
Using MINIMAT
Multiplicity
More Exercises
Real Similarity
DIFFERNTIAL EQUATIONS
Derivatives
Similarity and Differential Equation
Similarity and Powers
Using MINIMAT
Matrix Polynomials
Matrix Power Series
The Matrix Exponential
Using MINIMAT
The Companion Matrix
Using MINIMAT
HERMITIAN MATRICES
Hermitian Matrices Defined
Unitary Diagonalization
Using MINIMAT
Schur’s Theorem
Using MINIMAT
Spectral Theorem
Normal Spectral Theorem
Invariants
More Exercises
Real Normal Matrices
Positive Semidefinite Matrices
Skew-Hermitian matrices
Invariant subspaces
Conic Sections
TRIANGULAR MATRICES
Definitions
Factorization
Equivalence
The LU Decomposition
Uniqueness
Using MINIMAT
More Exercises
Back Substitution
Factorization Theorems
2 X 2 LU and Bruhat
Related Decompositions
Uniqueness of the LENF
Gershgorin’s Theorem
Real Triangular Equivalence
UNITARY MATRICES
Reflections
Using MINIMAT
Unitary Equivalence
Householder Decomposition
Using MINIMAT
Unitary Factorization
Using MINIMAT
Singular Values
Singular Value Decomposition
Invariants
More Exercises
Real Unitary Equivalence
Submultiplicative Norms
Polar Decomposition
Using MINIMAT
BLOCK DIAGONALIZATION
Generic Diagonalization
Monotriangular Block Diagonal Form (MTBDF)
Using MINIMAT
Nilpotent Matrices
Chevalley Decomposition
Using MINIMAT
More Exercises
Diagonalization
Generalized Eignenspaces
Matrix Exponential
Minimal Polynomial
Chevalley Decomposition
JORDAN NORMAL FORM
Similarity Invariants
Jordan Normal Form
Indecomposable Jordan Blocks
Partitions
Weyr Characteristic
Segre Characteristic
Jordan-Serge Basis
Improved Rank Nullity Relation
Proof of the Jordan Normal Form Theorem
More Exercises
Using MINIMAT
DETERMINANTS-II
Cofactors
The Companion Matrix
Adjoint
Cramer’s Rule
Using MINIMAT
Derivative of the Determinant
The Souriau-Frame Algorithm
A PROOFS
A.1 Matrix Algebra
A.2 Block Multiplication
A.3 The Fundamental Theorem
B MATHEMATICAL INDUCTION
C SUMMARY OF MINIMAT
C.1 Some Operations in MINIMAT
C.2 Columnwise Operations
C.3 Scalar Built-in Functions
C.4 Matrix Built-in Functions
C.5 Subscripts in MINIMAT
C.6 MINIMAT’s Entry wise Operations
C.7 Logical Operations
C.8 Control Structures
If, Elseif, Else
For
While
Break
Return
C.9 .M Functions Used in this Book
C.10 Miscellaneous Functions
C.11 Empty Matrices
D ANSWERS
E MINIMAT Tutorial (PC Version)
E.1 Before You Begin
E.2 Starting Up
E.3 The Prompt
E.4 Sample Session
E.5 Function Keys and Menus
E.6 Snow and Color
E.7 Transcript
E.8 Recall
E.9 Diary
E.10 SaveAs
E.11Viewing the Diary
E.12 Comments
E.13 Homework
E.14 Editing and Shell Escape
F INDEX
Biography
Joel W. Robbin
