Matrix algebra using MINImal MATlab

Cover
Half Title
Title Page
Copyright Page
Dedication
PREFACE
Table of Contents
1: WARMUP
	1.1 Clear Thinking
	1.2 Logic
	1.3 Proofs
	1.4 Sets
		Defining Sets by Enumeration
		Common Sets
		Sets and Properties
		Subsets
		Boolean Operations
		Equality of Sets
	1.5 Scalars
	1.6 Sigma Notation
	1.7 The Geometry of Linear Systems
	1.8 Using MINIMAT
		Assignment Statements
		Complex Numbers
		Expressions
		Scalar Built-in Functions
		. M Functions
		Format Command
		Control Structures
2: MATRIX OPERATIONS
	2.1 Matrices Defined
	2.2 Additive Operations on Matrices
	2.3 Multiplicative Operations
	2.4 Inverses
	2.5 Transpose
	2.6 Diagonal Matrices
	2.7 Triangular Matrice
	2.8 Matrices of Matrices
	2.9 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 Matrice
		Submatrices, Colon Notation
		Entrywise Operations
		Some Computer Exercises
	2.10 More Exercise
		Preview of the Exponential
		Geometric Series
3: INVERTIBLE MATRICES
	3.1 Elementary Row Operations
	3.2 Elementary Matrice
		Using MINIMAT
	3.3 Reduced Row Echelon Form
		Gauss-Jordan Elimination
		Computing the Multiplier
		Using MINIMAT
	3.4 How to Invert
		Using MINIMAT
	3.5 Elementary Column Operations
	3.6 Permutation Matrices
	3.7 Equivalence
		Using MINIMAT
4: SUBSPACES
	4.1 Linear Systems
		Using MINIMAT
	4.2 Null Space and Range
	4.3 Set Equality
		Using MINIMAT
	4.4 Subspaces
	4.5 New Subspaces from Old
	4.6 Bases
		Basis for the Null Space
		Basis for the Range
		Using MINIMAT
	4.7 Bases and Biequivalence
		Using MINIMAT
	4.8 The Range from the RREF
		Using MINIMAT
	4.9 Random Solutions
	4.10 Co-bases(*)
5: RANK AND DIMENSION
	5.1 The Definition of Dimension
		Using MINIMAT
	5.2 Existence
		Using MINIMAT
	5.3 The Analogy
	5.4 Rank and Nullity
		Using MINIMAT
	5.5 One-Sided Inverses
		Using MINIMAT
	5.6 Equivalence
	5.7 Uniqueness of the RREF
	5.8 More Exercises
		Characterizations of the Rank
		A Block Inverse Formula
		Geometry and Independence
		Matrix Representation on a Subspace
		Real Rank vs. Complex Rank
6: GEOMETRY
	6.1 Inner Products and Norms
		Real Inner Products
		Complex Inner Products
		Norms
	6.2 Geometric Interpretation
	6.3 Unitary Matrices
	6.4 Orthonormal Bases
	6.5 The Gram -Schmidt Decomposition
		Positive Triangular Matrices
		The Gram-Schmidt Process
		Geometric Interpretation
	6.6 Using MINIMAT
	6.7 Projection (General)
	6.8 Projection (Orthogonal)
		Using MINIMAT
	6.9 Least Squares
		The Best Approximate Solution
		The Closest Point
		Using MINIMAT
	6.10 More Exercises
		Submultiplicative Inequality
		Norms
		Pauli Matrices and Quaternions
7: DETERMINANTS-I
	7.1 Permutations
		Sign of a Permutation
		Transpositions
		Using MINIMAT
	7.2 Determinant Defined
		Easy Properties
		Computing Determinants
		Using MINIMAT
	7.3 More Exercises
		Wedge Product
		Real Equivalence
8: DIAGONALIZATION
	8.1 Similarity
	8.2 Eigenvalues and Eigenvectors
	8.3 Computing Eigenvalues
		Using MINIMAT
	8.4 The Characteristic Polynomial
		Using MINIMAT
	8.5 Multiplicity
	8.6 More Exercises
		Real Similarity
9: DIFFERENTIAL EQUATIONS
	9.1 Derivatives
	9.2 Similarity and Differential Equations
	9.3 Similarity and Powers
		Using MINIMAT
	9.4 Matrix Polynomials
	9.5 Matrix Power Series
	9.6 The Matrix Exponential
		Using MINIMAT
	9.7 The Companion Matrix
		Using MINIMAT
10: HERMITIAN MATRICES
	10.1 Hermitian Matrices Defined
	10.2 Unitary Diagonalization
		Using MINIMAT
	10.3 Schur’s Theorem
		Using MINIMAT
	10.4 Spectral Theorem
	10.5 Normal Spectral Theorem
	10.6 Invariants
	10.7 More Exercises
		Real Normal Matrices
		Positive Semidefinite Matrices
		Skew-Hermitian matrices
		Invariant subspaces
		Conic sections
11: TRIANGULAR MATRICES
	11.1 Definitions
	11.2 Factorization
	11.3 Equivalence
	11.4 The LU Decomposition
	11.5 Uniqueness
	11.6 Using MINIMAT
	11.7 More Exercises
		Back Substitution
		Factorization Theorems
		2 x 2 LU and Bruhat
		Related Decompositions
		Uniqueness of the LENF
		Gershgorin’s Theorem
		Real Triangular Equivalence
12: UNITARY MATRICES
	12.1 Reflections
		Using MINIMAT
	12.2 Unitary Equivalence
	12.3 Householder Decomposition
		Using MINIMAT
	12.4 Unitary Factorization
		Using MINIMAT
	12.5 Singular Values
	12.6 Singular Value Decomposition
	12.7 Invariants
	12.8 More Exercises
		Real Unitary Equivalence
		Submultiplicative Norms
		Polar Decomposition
		Using MINIMAT
13: BLOCK DIAGONALIZATION
	13.1 Generic Diagonalization
	13.2 Monotriangular Block Diagonal Form (MTBDF)
	13.3 Using MINIMAT
	13.4 Nilpotent Matrices
	13.5 Chevalley Decomposition
		Using MINIMAT
	13.6 More Exercises
		Diagonalization
		Generalized Eigenspaces
		Matrix Exponential
		Minimal Polynomial
		Chevalley Decomposition
14: JORDAN NORMAL FORM
	14.1 Similarity Invariants
	14.2 Jordan Normal Form
	14.3 Indecomposable Jordan Blocks
	14.4 Partitions
	14.5 Weyr Characteristic
	14.6 Segre Characteristic
	14.7 Jordan-Segre Basis
	14.8 Improved Rank Nullity Relation
	14.9 Proof of the Jordan Normal Form Theorem
	14.10 More Exercises
	14.11 Using MINIMAT
15: DETERMINANTS-II
	15.1 Cofactors
	15.2 The Companion Matrix
	15.3 Adjoint
	15.4 Cramer’s Rule
		Using MINIMAT
	15.5 Derivative of the Determinant
	15.6 The Souriau-Frame Algorithm
A: PROOFS
	A.1 Matrix Algebra
	A.2 Block Multiplication
	A.3 The Fundamental Theorem
B: MATHEMATIC ALINDUCTION
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 Entrywise 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.11 Viewing the Diary
	E.12 Comments
	E.13 Homework
	E.14 Editing and Shell Escape
F: INDEX