Index

J_k(λ), 182
k-th Moment, 197
m-form, 172

Adjugate of a Matrix, 51
Algebraic Multiplicity, 156
Annihilating Polynomial, 188

Basic Variables, 42
Basis Matrix, 89
Basis of a Vector Space, 80
Bilinear Form, 172

Cauchy-Schwartz Inequality, 121
Cayley Hamilton Theorem, 168
Change of Basis Matrix, 113
Characteristic Equation, 152
Characteristic Polynomial, 152
Derivative, 196
Characteristic Root, 151
Characteristic Value, 152
Characteristic Vector, 151
Characteristic-pair, 151
Closed Curve, 225
Cofactor Matrix, 51
Column vector, 7
Commuting Family of Matrices, 207
Companion Matrix, 188
Complex Vector Space, 60
Contour, 225
PICT PICT DRAFT Convergent Matrix, 211
convex hull, 114
Coordinate Matrix, 109
Coordinate Vector, 89
Curves, 225

Defective Matrix, 160
Definition
    Conjugate Transpose of a Matrix, 9
    Equality of two Matrices, 7
    Matrix, 7
    Transpose of a Matrix, 9
Deflation Technique, 158
Descartes’ Rule of Signs, 226
Determinant
    Properties, 220
Determinant of a Square Matrix, 49, 219
Diagonal matrix, 8
Diagonalizable Matrices, 159
Double Dual Space, 115
Dual Space, 115

Eigen-Condition, 151
Eigen-pair, 151
Eigen-space, 151
Eigenvalue, 151
Eigenvector, 151
Left, 154
Elementary Matrix, 31
Elementary Row Operations, 30
Elementary Symmetric Functions, 195
Equality of Linear Transformations, 95

Finite Dimensional Vector Space, 69
Form
    Hermitian, 172
    Linear, 173
    Quadratic, 172
    Sesquilinear, 172
Free Variables, 42
PICT PICT DRAFT Function
    bijective, 215
    injective, 215
    Inverse, 106
    Left Inverse, 106
    Multilinear, 172
    one-one, 215
    onto, 215
    Right Inverse, 106
    surjective, 215
Fundamental Theorem of Linear Algebra, 133

Geometric Multiplicity, 156
Given’s Method, 202
Gram-Schmidt Orthogonalization Process, 131

Hamel basis, 80
Hermitian Form, 172
Inertial degree, 174

Identity matrix, 8
Identity Operator, 95
Identity Transformation, 95
Index of an Eigenvalue, 183
Inertia of a matrix, 173
Inertial degree, 174
Inner Product, 119
Inner Product Space, 119
Invariant Subspace, 207
Inverse of a Linear Transformation, 106
Inverse of a Matrix, 14
Isometry, 144

Jacobi Method, 202
Jordan Block, 182
Jordan Canonical Form, 186
Jordan Curve, 225
Jordan Matrix, 182

PICT PICT DRAFT Latent Roots, 151
Leading Entry, 33
Left Eigenvector, 154
Linear Algebra
    Fundamental Theorem, 133
Linear Combination of Vectors, 67
Linear Dependence, 73
Linear forms, 173
Linear Functional, 114
linear Independence, 73
Linear Operator, 95
Linear Space
    Norm, 125
Linear Span of Vectors, 68
Linear System, 27
    Associated Homogeneous System, 28
    Augmented Matrix, 27
    Coefficient Matrix, 27
    Consistent, 28
    Equivalent Systems, 30
    Homogeneous, 27
    Inconsistent, 28
    Non-Homogeneous, 27
    Non-trivial Solution, 28
    Solution, 28
    Solution Set, 28
    Trivial Solution, 28
Linear Transformation, 95
    Composition, 112
    Equality, 95
    Inverse, 106, 112
    Isomorphism, 107
    Kernel, 98
    Matrix, 109
    Matrix Product, 112
    Non-Singular, 107
    Null Space, 98
    Nullity, 103
    Range Space, 98
    Rank, 103
    Scalar Product, 105
    Singular, 107
PICT PICT DRAFT     Sum, 105
Lower Triangular Matrix, 8

MAtrix
    Zero, 8
Matrix, 7
    Addition, 9
    Additive Identity, 10
    Additive Inverse, 10
    Adjugate, 51
    Change of Basis, 91, 113
    Cofactor, 51
    Column, 7
    Commuting Family, 207
    Companion, 188
    Defective, 160
    Determinant, 49
    Diagonal, 8
    Diagonalization, 159
    Eigen-pair, 151
    Eigenvalue, 151
    Eigenvector, 151
    Elementary, 31
    Generalized Inverse, 22
    Hermitian, 17
    Idempotent, 17
    Identity, 8
    Inverse, 14
    Jordan, 182
    Lower triangular, 8
    Minimal Polynomial, 188
    Negative definite, 170
    Negative semi-definite, 170
    Nilpotent, 17
    Non-derogatory, 160
    Non-Singular, 49
    Normal, 17
    Order, 7
    Orthogonal, 16, 139
    Permutation, 17, 213
    Positive definite, 170
    Positive semi-definite, 170
PICT PICT DRAFT     Principal Diagonal, 8
    Principal Minor, 195
    Principal Submatrix, 18
    Product of Matrices, 10
    Projection, 17
    Pseudo, 22
    Quadratic Form, 175
    Rank, 38
    Reflection, 17, 141
    Row, 7
    Row Echelon Form, 33
    Row Equivalence, 30
    Row-Reduced Echelon Form, 33
    Scalar, 8
    Scalar Multiplication, 9
    Singular, 49
    Size, 7
    Skew-Hermitian, 17
    Skew-Symmetric, 16
    Spectral Radius, 152
    Spectrum, 152
    Square, 8
    Submatrix, 18
    Symmetric, 16
    Toeplitz, 193
    Trace, 20
    triangular, 8
    Unitary, 17
    Upper triangular, 8
Matrix Equality, 7
Matrix Multiplication, 10
Matrix of a Linear Transformation, 109
Maximal linearly independent set, 79
Maximal subset, 79
Minimal Polynomial, 188
Multilinear function, 172

Non-derogatory Matrix, 160
Non-Singular Matrix, 49
Notation
    A(S∣T), 18
    A[S,T], 18
PICT PICT DRAFT     [k],k a positive integer, 18

Operator
    Identity, 95
    Self-Adjoint, 141
    Zero, 95
Order of Nilpotency, 17
Ordered Basis, 89
Orthogonal Complement, 124
Orthogonal Operators, 144
Orthogonal Projection, 138, 139
Orthogonal Vectors, 123
Orthogonally Congruent, 145
Orthonormal Basis, 127
Orthonormal Set, 127
Orthonormal Vectors, 127

Parseval’s formula, 130
Permutation, 215
    Cycle notation, 216
    Even, 219
    Odd, 219
Permutation Matrix, 17
Permutation:sgn function, 219
Permutation:Signature, 219
Pivot Entry, 33
Pivotal Column, 33
Plane Rotations, 199
Polynomial:Sign Changes, 226
Principal Minor, 195
Principal Submatrix, 18
prop:Schur:generalized, 181
Properties of Determinant, 220

QR Decomposition, 135
Generalized, 135
Quadratic Form, 172, 175

Rank of a Matrix, 38
PICT PICT DRAFT Rank-Nullity Theorem, 103
Real Vector Space, 60
Reflection Operator, 141
Reisz Representation Theorem, 98
Rigid Motion, 144
Row Equivalent Matrices, 30
Row Operations
Elementary, 30
Row vector, 7
Row-Reduced Echelon Form, 33

Scalar matrix, 8
Selection Principle of Unitary Matrices, 198
Self-Adjoint Operator, 141
Sesquilinear Form, 172
Signature of a matrix, 173
Similar Matrices, 113
Simple Closed Curve, 225
Simple Curve, 225
Simultaneously Diagonalizable, 206
Singular Matrix, 49
Solution Set of a Linear System, 28
Space
    Column Space, 72
    Linear, 60
    Normed Linear, 125
    Null Space, 72
    Range, 72
    Row Space, 72
Spectrum of a matrix, 152
Square Matrix
    Bilinear Form, 172
    Determinant, 219
Square matrix, 8
Standard Basis for ℂⁿ, 8
Star congruence, 174
Submatrix of a Matrix, 18
Subset
    Maximal, 79
    Maximal linear independent, 79
    Sum, 70
Subspace
PICT PICT DRAFT     Linear Span, 70
    Orthogonal Complement, 123
    Sum, 70
Sum of Principal Minors, 195
Sum of two Matrices, 9
Sylvester’s law of inertia, 174
System of Linear Equations, 27

Toeplitz Matrix, 193
Trace of a Matrix, 20
Transformation
Zero, 95
Triangular Matrix, 8
Trivial Subspace, 65

Unit Vector, 17, 121
Unitarily Congruent, 145
Unitary Equivalence, 163
Unitary Group, 198
Unitary Similar, 163
Upper triangular form, 8
Upper triangular matrix, 8

Vector
    Column, 7
    Coordinate, 89
    Row, 7
    Unit, 17
Vector Space, 60
    Basis, 80
    Complex, 60
    Complex n-tuple, 62
    Dimension of M + N, 223
    Dual Basis, 115
    Finite Dimensional, 69
    Infinite Dimensional, 69
    Inner Product, 119
    Isomorphic, 107
    Minimal spanning set, 80
    Real, 60
PICT PICT DRAFT     Real n-tuple, 62
    Subspace, 64
Vector Subspace, 64
Vectors
    Angle, 122
    Length, 121
    Linear Combination, 67
    Linear Dependence, 73
    Linear Independence, 73
    Linear Span, 68
    Mutually Orthogonal, 127
    Norm, 121
    Orthogonal, 123
    Orthonormal, 127

Word in Symbols s and t, 205

Zero matrix, 8
Zero Operator, 95
Zero Transformation, 95