# Explanation regarding eigenvectors and eigenvalues

In PCA algorithm, $$C$$ the covariance matrix is defined as $$\frac1{N-1}XX^T$$, where $$X$$ is a matrix of size $$d\times n$$. Also, the rank of $$X$$ is $$\min(d-1,n)$$. (Lets assume)
Now, since $$C$$ is symmetric and PSD, all its eigenvalues are real and non negative.
We know that number of eigenvalues of a $$d \times d$$ matrix is $$d$$.
Now my question(s) are,
(i) how do we know that there are exactly, $$r=\min(d,n-1)$$ linearly independent eigenvectors corresponding to $$r$$ non-zero eigenvalues?
(ii) and that the rest $$0$$ eigenvalues correspond to $$(d-r)$$, linearly independent eigenvectors.
(iii) and also that even together, all these eigenvectors are linearly independent. ((iv) are they all orthogonal).
I am saying this because in PCA algorithm, $$CV=V\Lambda$$ where $$V$$ is $$d\times d$$ eigenvector (at columns) matrix, which is orthogonal, and $$\Lambda$$ is a $$d\times d$$ eigenvalue matrix.

Note that to prove the claims, I cannot use SVD, since it is introduced later, and is derived using the facts I wrote above (in some way).