1, any principal matrix of a positive definite matrix is also a positive definite matrix.

2. If A is a symmetric positive definite matrix of order n, there exists a unique lower triangular matrix L with positive principal diagonal elements, so that A = L * L'. This decomposition formula is called Coleski decomposition of positive definite matrix.

3. If A is a positive definite matrix of order N, then A is a reversible matrix of order N. ..

Second, determine the semi-positive definite matrix:

1. For semi-positive definite matrices, the corresponding condition should be changed to that all principal components and principal components are nonnegative. The non-negative principal component of a sequence does not mean that the matrix is semi-positive.

2. Semi-positive definite matrix: Let A be a real symmetric matrix. If any real nonzero column matrix x has XT*A*X≥0, then A is called a semi-positive definite matrix.

3.A∈Mn(K) is a semi-positive definite matrix if all the principal components of A are greater than or equal to zero.

Third, the judgment of negative definite matrix:

1, let a be a real symmetric matrix. If there is xtax,

2.A∈Mn(K) is a positive definite matrix if and only if -A is a positive definite matrix.

3.A∈Mn(K) is a positive definite matrix if and only if $ A {- 1} $ is a positive definite matrix.

4.A∈Mn(K) is a negative definite matrix if and only if all odd-numbered principal components of A are less than zero and all even-numbered principal components are greater than zero.

Extended data:

positive definite

If the real symmetric matrix A of n×n meets the requirements of all non-zero vectors

The corresponding quadratic form:

If Q>0 is called a positive definite matrix. If q; =0, a is a semi-positive definite matrix; If a is neither semi-positive nor semi-negative, a is an indefinite matrix? . The positive definiteness of a symmetric matrix is closely related to its eigenvalue. A matrix is positive definite if and only if its eigenvalues are all positive numbers.

The real symmetric matrix A is negative definite, if the quadratic form f(x 1, x2, ..., xn)=X'AX is negative definite. The necessary and sufficient condition for a matrix to be negative is that its eigenvalues are all less than zero. If the matrix A is a negative definite matrix of order n, then the even-order principal component of A is greater than 0, and the odd-order principal component is less than 0.

The real symmetric matrix a is called semi-positive definite. If the quadratic form X'AX is semi-positive, that is, for any real column vector x that is not 0, x' ax ≥ 0;

References:

Baidu Encyclopedia-Matrix

References:

Baidu encyclopedia-semi-positive definite matrix

References:

Baidu encyclopedia-negative definite matrix