Show that for a symmetric positive-definite matrix we can write $A = D B D$ [closed]

Q

Let A be a symmetric positive-definite $m \times m$ matrix.

Show we can write $A = D B D$, where $D$ is diagonal and $B$ has only ones on the diagonal.

A

To be honest... Not sure where to start with this one... Specifically, I'm not sure how to get ones along the diagonal of $B$. I'm guessing that $D$ is going to end up being the diagonal matrix of singular values of $A$.

I'm also trying to figure out what the significant of having all 1's along the diagonal is.

Solution 1:

Let $a_{ii}$ be the diagonal entries of $A$. Then since $A$ is positive definite, we have that $a_{ii} > 0$. ($a_{ii} = e_iAe_i > 0$).

So let $d_{ii} = \sqrt{a_{ii}}$ and let $B = D^{-1}AD^{1}$. Notice that $B$ has ones on the diagonal.