Matrix Decomposition / Factorization 相关重点 (特征值特征向量)

General but Important

Given a matrix, ask yourself,

• is it square matrix?
• is it symmetric? eigenvectors are orthonormal for distinct eigenvalues
• can be decomposed? Eigendecomposed? SVD?
• How about is eigenvalues and eigenvectors? is it psd?
• can it be diagonalized?

Eigen Values, Eigen Vectors, Eigen Space

I have listed some important points

• Thus, for example, the exponential function f(x)=eλx is an eigenfunction of the derivative operator ′ , with eigenvalue λ , since its derivative is f′(x)=λeλx=λf(x)

• An eigenbasis for A is any basis for the set of all vectors that consists of linearly independent eigenvectors of A . Not every matrix has an eigenbasis, but every symmetric matrix does

• The prefix eigen- is adopted from the German word eigen for “own-“

• Remember some properties

  • The trace of A, defined as the sum of its diagonal elements, is also the sum of all eigenvalues:

  • The determinant of A is the product of all eigenvalues

  • The eigenvalues of the kth power of A, i.e. the eigenvalues of Ak , for any positive integer k, are λk1,λk2,…,λkn

• The set of eigenvalues of T is sometimes called the spectrum of T

Diagonalizable Matrix 对角化

• Definition: for a n × n matrix, i.e. square matrix, PAP−1 is a diagonal matrix, or A can be formulated as A=PXP−1 , where X is a diagonal matrix.

• An n×n matrix A is diagonalizable over the field F if it has n distinct eigenvalues in F. however, the converse may be false

• See Diagonalization Prove in wiki. why the diagonalization result is related to the eigenvectors and eigenvalues.

• if square matrix A is symmetric, eigenvectors of A can be formed an orthonormal basis, so P is orthogonal matrix. This can be called eigen decompostion of such matrix

• used in calculating the power of a matrix

Positive Semi-definite Matrix

https://en.wikipedia.org/wiki/Positive-definite_matrix

Eigen Decomposition (Spectral Decomposition)

refer to the diagonalization
https://en.wikipedia.org/wiki/Eigendecomposition_of_a_matrix

Singular Value Decomposition (SVD)

• For m × n matrix A, there exists SVD !!!!!

• U and V are orthonormal matrix

Square Root of a Matrix

https://en.wikipedia.org/wiki/Square_root_of_a_matrix

Gram matrix

see the wiki

Others

• Hermitian matrix is the extension for the sysmetric matrix

• Unitary matrix is the extension for the orthogonal matrix

Ref

-https://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors

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