Relation between rank of a symmetric positive semi-definite matrix and its number of non-zero eigen values (or singular values)

Is there any relation between the rank of a symmetric positive semi-definite matrix and its number of non-zero eigenvalues (or singular values)?

For a matrix $\mathbb{P}$ Can we find the relationship between the rank and the eigenspace of the matrix?

• The "multiplicity of the zero eigenvalues" is always 1, right? Because 0 would be the only eigenvalue of the nullity(A) (the nullspace of $A$). Aug 28 '17 at 16:03
• @Sother it is common to include symmetry as part of the definition of "positive definite", but I agree I should reword this answer. The multiplicity is not always $1$; for example, $$\pmatrix{1&1&1\\1&1&1\\1&1&1}$$ has a $0$-eigenvalue of multiplicity $2$. Aug 28 '17 at 19:30