Definition
Judge the definiteness of matrix M
- When M is not symmetric:
We first build a new matrix: 0.5(M+M'). It should be symmetric. Then we judge whether this matrix is positive definite.
- When M is symmetric:
Positive definite:
Approach 1: Use leading principal minor (LPM) and principal minors (PM). Here LPM and PM are the determinants of associated matrix.
All LPM > 0, from k = 1 to n (n: dimension of the matrix M)
Or:
Approach 2: Use eigen values
Z'MZ > 0 for any Z is non-zero matrix.
Positive Semi-Definite:
All PM 
0, from k = 1 to n
Or:
Z'MZ 
0 for any Z is non-zero matrix.
Negative Definite:
All 
LPM > 0, k is the order of LPM. (k is odd, LPM < 0; k is even, LPM > 0)
Or:
Z'MZ < 0 for any Z is non-zero matrix.
Negative Semi-definite:
All PM 
0, k is the order of PM. (k is odd, PM 
0; k is even, PM 
0)
Or:
Z'MZ 
0 for any Z is non-zero matrix.
Note
- M': transpose of M