4.4 The Schur Decomposition

• Contents
• 1. Linear equations and elementary matrix algebra - a review
  • 1.1 Gaussian Elimination
  • 1.2 Vectors in ⁿ and ⁿ
  • 1.3 Matrix Operations
  • 1.4 Matrix Inverses
  • 1.5 The LU Decomposition
• 2. Vector Spaces and Linear Transformations
  • 2.1 A Model for General Vector Spaces
  • 2.2 The Basics of Bases
    • 2.2.1 Spanning Sets and Linear Independence
    • 2.2.2 Basis and Dimension
    • 2.2.3 Change of Basis
  • 2.3 Linear Transformations and their Representation
    • 2.3.1 Matrix Representations
    • 2.3.2 Similarity of Matrices
  • 2.4 Determinants
• 3. Inner Products and Best Approximations
  • 3.1 Inner Products
  • 3.2 Best approximation and projections
  • 3.3 Pseudoinverses
  • 3.4 Orthonormal Bases and the QR Decomposition
  • 3.5 Unitary Transformations and the Singular Value Decomposition
• 4. The Eigenvalue Problem
  • 4.1 Eigenvalues and Eigenvectors
    • 4.1.1 Eigenvalue Basics
    • 4.1.2 The Minimal Polynomial
  • 4.2 Invariant Subspaces and Jordan Forms
    • 4.2.1 Invariant Subspaces
    • 4.2.2 Jordan Forms
  • 4.3 Diagonalization
  • 4.4 The Schur Decomposition
  • 4.5 Hermitian and other Normal Matrices
    • 4.5.1 Hermitian matrices
    • 4.5.2 Normal Matrices
    • 4.5.3 Positive Definite Matrices
    • 4.5.4 Revisiting the Singular Value Decomposition
• Index

Prerequisites:

• Basic definitions for the eigenvalue problem
• Matrix algebra.
• Unitary matrices and orthogonal bases

Learning Objectives:

• Familiarity with the construction of unitary triangularization.
• Understanding in what sense nondiagonalizable matrices are always close to dagonalizable matrices.

Index

augmented matrix : 1.1

$\mathbb {C}$ⁿ : 1.2

dot product : 1.2

Euclidean norm : 1.2

diagonal : 1.4

elementary row operations : 1.1

elementary transformations : 1.1

Gauss elimination : 1.1

Gauss transformation : 1.5

Gauss-Jordan elimination : 1.1

identity matrix : 1.3

invertible : 1.4

kernel : 2.1

lower triangular : 1.5

n-vector : 2.1

permutation matrix : 1.5

permuted LU decomposition : 1.5

range : 2.1

reduced row echelon form : 1.1

$\mathbb {R}$ⁿ : 1.2

dot product : 1.2

Euclidean norm : 1.2

vectors : 1.2

angle : 1.2

distance : 1.2

open ball : 1.2

scalar multiplication : 1.2

vector addition : 1.2

singular : 1.4

subspace : 2.1

system of linear equations : 1.1

consistent : 1.1

homogeneous : 1.1

inconsistent : 1.1

nonhomogeneous : 1.1

solution : 1.1

trivial : 1.1

solution set : 1.1

transpose : 1.3

upper triangular : 1.5

vector space : 2.1