I have the matrix $A$ which is of size $m \times n$, with $m > n$. Let $Q = A^T A$.
How do I determine, if $Q$ is Semi-Positive-definite?
I have the matrix $A$ which is of size $m \times n$, with $m > n$. Let $Q = A^T A$.
How do I determine, if $Q$ is Semi-Positive-definite?
Recall that $B$ is semi-positive-definite iff $x^T B x\ge 0$ for all vectors $x$, where the superscript $T$ denotes the transpose. In this case, $x^TQx=x^T A^T A x.$ Now, recall that $(Bv)^T=v^T B^T$, so $x^TQ x=(Ax)^T(Ax).$ Finally, note that the square of the norm $\|v\|$ of a vector $v$ is just $\|v\|^2=v^T v$, so $x^T Qx=\|Ax\|^2 \ge0,$ and this gives you that $Q$ is as wanted.
Note that in general this won't be positive definite, because there may be values of $x\ne0$ such that $Ax=0$. (This would be guaranteed if $m