Regarding trace of idempotent matrix multiplied by its transpose

Suppose $M$ is a $n$ by $n$ idempotent real matrix of rank $p$, and therefore of trace $p$. Is there anything obvious we can say about the trace of $M^TM$ in terms of $p$? Note $M$ is not necessarily an orthogonal projection so $M \neq M^T$ in general.

Solution 1:

If $p=0$ and $p=n$ this is trivial, since there is only one idempotent of rank $p$.

Claim: Let $1\le p< n$ be integers. The range of $M\longmapsto\mathrm{Tr}(M^TM)$ is equal to $[p,+\infty)$ when $M$ runs over the set of all $n\times n$ real idempotents of rank $p$.

Sketch of proof:

• By continuity and connectedness, the range is an interval.
• Easy examples show that the range contains $[p,+\infty)$.
• Given an arbitrary idempotent $M$ of rank $p$, the characteristic polynomial of $M$ is $(X-1)^pX^{n-p}$. Since it splits over $\mathbb{R}$, we can find an orthogonal matrix $P$ such that $N=P^TMP$ is upper triangular with a diagonal made of $n-p$ zeros and $p$ ones. It follows that $$\mathrm{Tr}(M^TM)=\mathrm{Tr}(N^TN)\ge 1^2+\cdots+1^2=p$$ and the proof is complete.