Is Positive Semi-Definiteness of a Matrix a loose measure of Independence?

Question

I am trying to understand D-optimality criteria which tends to minimize the covariance matrix between two random variables. Now keeping that aside and in general, having PSD of a covariance matrix imply that the random variables are linearly independent?

Also for the reference I found this answer

determinant=Product of eigenvalues and a positive definite matrix have all positive (strictly greater than zero) eigenvalues.

So does having an eigen value greater then 0 imply that there is a correlation and if so does it make another loose measure of independence?

Answer 1

having PSD of a covariance matrix imply that the random variables are linearly independent

No, all covariance matrices are PSD.

So does having an eigen value greater then 0 imply that there is a correlation and if so does it make another loose measure of independence?

If you have all eigenvalues greater than zero, then the covariance matrix will be PD. And you will be able to use a whitening transformation to transform your original random vector $X$ into a $Y$ that is uncorrelated. Just having uncorrelatedness is not enough for you to claim they are independent, in general.

But, if you are dealing with a random vector $X$ with a multivariate normal distribution, you can conclude that after the transformation the components of $Y$ are not only uncorrelated, but also independent.