Positive definite matrices

A kind of self-conjugate matrix of positive definite matrices. Positive definite matrices are similar to positive real numbers in complex numbers. Definition: For symmetric matrix M, when and only if there is any vector x, there is

If the upper formula is greater than or equal to zero, then M is the semi positive definite matrix. The positive definite matrix is recorded as m>0.

Also known as positive definite two-time

Determination of positive definite matrices

1, all eigenvalues are positive (according to the spectral theorem, if the condition is established, it is necessary to find the diagonal matrix D and positive definite matrix P, so M=P^-1DP);
2, all the order of the principal type is positive definite;
3, Cholesky decomposition of the matrix, its main diagonal elements are all positive;
4, the matrix has a semi-bilinear mapping form.

The bilinear mapping is explained first. Suppose three vector spaces X, y and z, have z = B (x, y). There is a unique mapping to Z for any vector in X or Y. If x is fixed, the elements in Y are linearly mapped to Z, and vice versa.
The so-called semi-bilinear mapping, is that its two parameters one is linear, the other is a semi-linear (or conjugate linear). Such as:

The inner product of the complex space is semi-bilinear.

Properties of positive definite matrices

1, positive definite matrices are reversible, and inverse matrices are positive definite matrices;
2, positive definite matrix and positive real number product also is positive definite;
3. Trace TR (M) >0;
4, there is a unique square root matrix B, so that:

Positive definite matrices