Is this Matrix Invertible? Suppose $X$ is a real $n\times n$ matrix. Suppose $m>0$ and let $\operatorname{tr}(X)$ denote the trace of $X$. If $\operatorname {tr}(X^{\top}X)=m$, can i conclude that $X$ is invertible?
Thanks 

No, consider 
$X=\pmatrix{
1 & 0 \\
0 & 0
}
$. Then $\operatorname{tr}(X^tX)=1$ but $X$ is not invertible.
