# Property of Covariance Matrices and Symmetric Matrices

I have a question about covariance matrices. I have read one interesting property that, all symmetric matrices are diagonalizable.

Suppose we have a data matrix $$X$$ that has only $$m$$ independent columns. Additionally, the corresponding covariance matrix $$\Sigma = (X - \mu)^T(X -\mu)$$

My question is, say the covariance matrix, $$\Sigma$$, lives in $$R^{p *p}$$, but this matrix will only have $$m$$ independent columns. Does this imply that we still get $$p$$ (full) orthogonal eigenvectors or only $$m$$ orthogonal eigenvectors and the remaining $$p - m$$ independent but not orthogonal eigenvectors?

• For a symmetric matrix with $p$ rows and columns, you can always find $p$ orthogonal eigenvalues see here. – John L Mar 13 at 14:22
• For some intuition, consider the case $m=0.$ – whuber Mar 13 at 18:54