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Chapter 8: Orthogonality

Chapter 8: Orthogonality

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Chapter 8: Orthogonality
8.1 Orthogonal Complements and Projections
Lemma 8.1.1: Orthogonal Lemma
Theorem 8.1.1
Proof.
Theorem 8.1.2: Gram-Schmidt Orthogonalization Algorithm1
Example 8.1.1
Remark
Projections
Definition 8.1 Orthogonal Complement of a Subspace of Rn
Lemma 8.1.2
Proof.
Example 8.1.2
Definition 8.2 Projection onto a Subspace of Rn
Theorem 8.1.3: Projection Theorem
Proof.
Example 8.1.3
Example 8.1.4
Theorem 8.1.4
Exercises for 8.1
Exercise 8.1.6
8.2 Orthogonal Diagonalization
Theorem 8.2.1
Definition 8.3 Orthogonal Matrices
Example 8.2.1
Example 8.2.2
Example 8.2.3
Definition 8.4 Orthogonally Diagonalizable Matrices
Theorem 8.2.2: Principal Axes Theorem
Example 8.2.4
Theorem 8.2.3
Theorem 8.2.4
Example 8.2.5
Example 8.2.6
Theorem 8.2.5: Triangulation Theorem
Corollary 8.2.1
Exercises for 8.2
8.3 Positive Definite Matrices
Definition 8.5 Positive Definite Matrices
Theorem 8.3.1
Theorem 8.3.2
Example 8.3.1
Example 8.3.2
Lemma 8.3.1
Theorem 8.3.3
Algorithm for the Cholesky Factorization
Example 8.3.3
Exercises for 8.3
Exercise 8.3.2
Exercise 8.3.12
8.4 QR-Factorization
Definition 8.6 QR-factorization
Theorem 8.4.1: QR-Factorization
Example 8.4.1
Corollary 8.4.1
Theorem 8.4.2
Remark
Theorem 8.4.3
Exercises for 8.4
8.5 Computing Eigenvalues
The Power Method
Example 8.5.1
QR-Algorithm
Example 8.5.2
Exercises for 8.5
8.6 The Singular Value Decomposition
Lemma 8.6.1
8.6.1 Singular Value Decompositions
Lemma 8.6.2
Proof.
Definition 8.7
Definition 8.8
Theorem 8.6.1
Example 8.6.1
SVD Algorithm
Definition 8.9
Lemma 8.6.3
8.6.2 Fundamental Subspaces
Definition 8.10
Lemma 8.6.4
Proof.
Theorem 8.6.2
Example 8.6.2
8.6.3 The Polar Decomposition of a Real Square Matrix
Definition 8.11
Lemma 8.6.5
Proof.
Definition 8.12
Theorem 8.6.3
8.6.4 The Pseudoinverse of a Matrix
Theorem 8.6.4: Penrose’ Theorem12
Definition 8.13
Theorem 8.6.5
Definition 8.14
Lemma 8.6.6
Theorem 8.6.6
Example 8.6.3
Lemma 8.6.7
Exercises for 8.6
8.7 Complex Matrices
The Standard Inner Product
Definition 8.15 Standard Inner Product in Rn
Example 8.7.1
Theorem 8.7.1
Definition 8.16 Norm and Length in Cn
Theorem 8.7.2
Example 8.7.2
Definition 8.17 Conjugate Transpose in Cn
Example 8.7.3
Theorem 8.7.3
Hermitian and Unitary Matrices
Definition 8.18 Hermitian Matrices
Example 8.7.4
Theorem 8.7.4
Theorem 8.7.5
Definition 8.19 Orthogonal and Orthonormal Vectors in Cn
Theorem 8.7.6
Definition 8.20 Unitary Matrices
Example 8.7.5
Example 8.7.6
Unitary Diagonalization
Theorem 8.7.7: Schur’s Theorem
Corollary 8.7.1
Theorem 8.7.8: Spectral Theorem
Example 8.7.7
Theorem 8.7.9
Theorem 8.7.10: Cayley-Hamilton Theorem17
Exercises for 8.7
Exercise 8.7.10
Exercise 8.7.11
Exercise 8.7.12
Exercise 8.7.16
Exercise 8.7.18
Exercise 8.7.21
8.8 An Application to Linear Codes over Finite Fields
Modular Arithmetic
Finite Fields
Theorem 8.8.1
Example 8.8.1
Example 8.8.2
Example 8.8.3
Error Correcting Codes
Theorem 8.8.2
Nearest Neighbour Decoding
Theorem 8.8.3
Proof.
Example 8.8.4
Theorem 8.8.4: Hamming Bound
Linear Codes
Example 8.8.5
Theorem 8.8.5
Proof.
Example 8.8.6
Matrix Generators
Lemma 8.8.1
Parity-Check Matrices
Theorem 8.8.6
Orthogonal Codes
Theorem 8.8.7
Exercises for 8.8
Exercise 8.8.11
Exercise 8.8.12
Exercise 8.8.13
8.9 An Application to Quadratic Forms
Definition 8.21 Quadratic Form
Example 8.9.1
Theorem 8.9.1: Diagonalization Theorem
Example 8.9.2
Theorem 8.9.2
Example 8.9.3
Solution.
Congruence
Theorem 8.9.3
Example 8.9.4
Theorem 8.9.4: Sylvester’s Law of Inertia
Example 8.9.5
Theorem 8.9.5
Proof.
Proof of Theorem 8.9.4.
Exercises for 8.9
8.10 An Application to Constrained Optimization
Example 8.10.1
Theorem 8.10.1
Example 8.10.2
Example 8.10.3
8.11 An Application to Statistical Principal Component Analysis