Is there any relationship between highest eigenvalue and rank of the PSD matrix? Let's say there are two PSD matrices $A$ and $B$ with rank $r_A$ and $r_B$ respectively and their traces are equal. Let's say that $\lambda_A$ and $\lambda_B$ are the highest eigenvalue of $A$ and $B$ respectively. Then can we say the following?
If $ r_A \geq r_B $ then $ \lambda_A \leq \lambda_B $?
The answer is no. The rank of a PSD matrix counts the number of non-zero eigenvalues (with multiplicity).
The answer is still no. Consider $$ \pmatrix{4 \\&1\\&&1}, \qquad \pmatrix{3\\&3\\&&0} $$