I was looking at UCLA's Department of Mathematics's Fall 2015 Basic Qualifying Exam, and question #12 is the following.
Show that the following matrix is positive definite.
$$M=\begin{pmatrix}
2 & 1 & 1 & \ldots & 1\\
1 & 3 & 1 & \ldots & 1\\
1 & 1 & 4 & \ldots & 1\\
\vdots & \vdots & \vdots & \ddots & \vdots\\
1 & 1& 1& \ldots &n+1
\end{pmatrix}$$
I wasn't super sure how one should proceed; showing $x^TMx>0$ for all $x \neq 0$ seems like a pain, and I was curious if there was a slick way to do this. Other ways would be to write $M = AA^T$ or go through the characteristic polynomial, but I'm not super sure about those either. If anyone can give some hints, that would be great!
Best Answer
Write $$M=\begin{pmatrix} 1 & 0 & 0 & \ldots & 0\\ 0 & 2 & 0 & \ldots & 0\\ 0 & 0 & 3 & \ldots & 0\\ \vdots & \vdots & \vdots & \ddots & \vdots\\ 0 & 0& 0& \ldots &n \end{pmatrix}+ \begin{pmatrix} 1 & 1 & 1 & \ldots & 1\\ 1 & 1 & 1 & \ldots & 1\\ 1 & 1 & 1 & \ldots & 1\\ \vdots & \vdots & \vdots & \ddots & \vdots\\ 1 & 1& 1& \ldots &1 \end{pmatrix}.$$
Now it's easy to see that the first of these matrices is positive-definite, while the second is positive-semidefinite (in particular, its eigenvalues are $n$ and $0$).