Create Time: 23th Dec 2023
Title: "[[Eigenvalues-and-Eigenvectors]]"
status: DONE
Author:
- AllenYGY
tags:
- Eigenvectors
- Eigenvalues
- Lec4
- NOTE
- LinearAlgebra
created: 2023-12-23T02:12
updated: 2024-04-08T22:58
Find eigenvalue and eigenvector
Property:
Let A be an n×n matrix and
If
An n×n matrix A is said to be diagonalizable if there exists a nonsingular matrix X and a diagonal matrix D such that
An n×n matrix A is diagonalizable if and only if A has n linearly independent eigenvectors.
trace = a+d =
det(A)=
A square matrix A is orthogonally diagonalizable if
If A is a real symmetric matrix, then there is an orthogonal matrix Q that diagonalizes A; that is,
Properties
Suppose a m×n matrix A Then
has m orthonormal eigenvectors
has n orthonormal eigenvectors
Assume that A is an m×n matrix with m ≥ n.
Factorize A into a product
If A is an m × n matrix, then A has a singular value decomposition.
2×2:
A real n×n symmetric matrix A is said to be
If A is a real symmetric n×n matrix, then there is a change of variables
Let A be a real symmetric n×n matrix. Then A is positive definite if and only if all its eigenvalues are positive.
A is
Let A be a real symmetric n×n matrix. The following are equivalent:
Leading principal submatrices:upper-left 1×1, 2×2, 3×3,…
If some determinants are positive, and others negative: A is indefinite.
If these determinants are all ≥0: positive semidefinite.