Are not all symmetric invertible matrices positive semidefinite??

Hi,

I remember to learn that a symmetric matrix is positive semidefinite if and only if it is invertible. But the matrix 'covmat' in the .mat file that you can download using the below link is symmetric, invertible, but not positive semidefinite.

http://homes.cs.washington.edu/~safiye/ask.mat

After loading ask.mat, you can try below code portions to check what I am saying.

issym(covmat); % you will get answer 1
inv(covmat); % you will get no error
mvnrnd(zeros(110, 1), covmat, 20); % you will get the error below:

??? Error using ==> mvnrnd at 118
SIGMA must be a symmetric positive semi-definite matrix.

How can it be possible? What is wrong with this matrix? Do I have some wrong knowledge of matrix properties of invertibleness, positive semidefiniteness, etc.?

Thanks!

Answer 1

[the cyclist]

Your memory is incorrect. Consider the following matrix:

a = [ 1.0 0.8 -0.8;
      0.8 1.0  0.8;
     -0.8 0.8  .01]

It is not positive semi-definite (because it has a negative eigenvalue). However, it is symmetric and invertible. (It has non-zero determinant).