# A question about the symmetric positive definite matrix A and $D^{-1/2}AD^{-1/2}$

Assume that $$A\in\mathbb{R}^{n\times n}$$ is a symmetric positive definite matrix and $$D=diag(d_1,\ldots,d_n)$$ is a diagonal matrix constructing by using the diagonal entries of $$A$$, indeed $$d_i=a_{ii}$$.

What can we say about the relationship between the eigenvalues and eigenvectors of $$A$$ and the eigenvalues and eigenvectors of $$D^{-1/2}AD^{-1/2}$$?

$$D^{-1/2} A D^{-1/2}$$ is positive definite. It has diagonal entries $$1$$, so the sum of its eigenvalues is its trace, which is $$n$$. The trace of $$A$$ could be any positive number.