[Math] For a covariance matrix, what would be the properties associated with the eigenvectors space of this matrix

Question

I want to know, since the covariance matrix is symmetric, positive, and semi-definite, then if I calculate its eigenvectors what would be the properties of the space constructed by those eigenvectors (corresponds to non-close-zero eigenvalues), is it orthogonal or anything else special?
Suppose this eigenvector matrix is called U, then what would be the properties with
U*transpose(U)？

Answer 1

A symmetric matrix has orthogonal eigenvectors (irrespective of being positive definite - or zero eigenvalues). Hence, if we normalize the eigenvectors, U * transpose(U) = I