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
- 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
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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