Methods for Definiteness of a matrix are giving different results

Question

I've found several different sets of rules to check for definitness. They are:

1. Determinant of all upper-left submatrices, shown here

2. Check the sign of the eigenvalues of A, which is the definition given in my course notes.

3. Check the sign of the eigenvalues of the hermitian part of A, like the answer shown here

The problem is, each of these definitions appear to give me different results some times. (i.e., there are times when one will tell me the matrix is Positive Definite, while another tells me it is Indefinite).

I have checked my implementation repeatedly but I can't quite figure out what the problem is. I'm aware that some definitions of definiteness require the matrix be symmetric, but as far as I'm aware I am not using that definition for my course.

For example, the matrix:

$A = \begin{bmatrix} 489.0740 & 47.9540 & 223.9514\\ 413.2403 & 60.4261 & 23.0003\\ 291.3468 & 418.2741 & 176.7738 \end{bmatrix}$

has the determinants of all of its upper left submatrices being:

$D = [489.0740, \ 9.7363e+03, \ 3.2104e+07]$

Which should indicate that the matrix is positive definite (since they are all positive). However the eigenvalues of A are found to be

$eigenvalues \ = \ 1.0e+02 * [7.4150 + 0.0000i, \ -0.0761 + 2.0794i, \ -0.0761 - 2.0794i]$

And the eigenvalues of the hermitian part (given by $A_H = \frac{1}{2}(A + A^H)$), are found to be:

$eigenvalues \ = \ [-111.4704, \ 67.2772, \ 770.4671]$

Both of which indicate the matrix should be indefinite.

Alternatively, the matrix given by:

$A = \begin{bmatrix} 395.3073 & 130.0201 & 0.2004\\ 496.9838 & 492.5484 & 486.9109\\ 39.6018 & 180.3315 & 366.8912 \end{bmatrix}$

The determinants of the upper-left submatrices come out to be:

$D = [395.3073, \ 1.3009e+05, \ 1.5540e+07]$

and the eigen values of A are found to be:

$eigenvalues \ = \ [837.4463, \ 366.6964, \ 50.6042]$

Both of which indicate that A should be positive definite. However, looking at the eigenvalues of the hermitian part gives us:

$eigenvalues \ = \ [-14.1688, \ 362.2705, \ 906.6452]$

Which is clearly indefinite.

What am I missing here? I'm clearly doing something wrong, or misunderstanding something... but I cannot for the life of me figure out what it is. The only notes from class state that all you need to check are signs of the eigenvalues of $A$. However from what I've read online, that shouldn't be giving me different results from these other methods.

Answer 1

Several of your tests (such as Sylvester's Criterion, or positivity of the eigenvalues) apply only to Hermitian matrices. Your example matrix is not Hermitian.

For a general matrix, if your definition of definiteness is that $x^HMx > 0$ for all nonzero vectors $x$, then the condition you need to check is that the Hermitian part of $M$ is positive-definite. You can apply your various tests (Sylvester's Criterion etc) to this Hermitian matrix (and not the original $M$).