Given SPD matrix $$A \in \mathbb{R}^{N \times N} $$ and positive diagonal matrix $$D \in \mathbb{R}^{N\times N}.$$ What is then spectrum of the product $$D^TAD.$$ Is there a closed-form relationship between spectra of $A, D$ and that of the product? Or at least can we find bounds of it?E.g., one can see that if $D = \alpha I$ then $cond(A) = cond(D^TAD)$ for any $\alpha > 0$. UPDATE: In practice I observe that for arbitrary positive $D$ the $cond(A) < cond(D^TAD)$. However, I can not explain or understand it. Any clues are appreciated.UPDATE:The following MATLAB script generates $k$ random matrices $A$ and $D$ and then $i$ times decreases the smallest diagonal element of $D$ by factor $r < 1$, calculating at each iteration condition number of the product $D^TAD$ with updated $D$:sz = 100;v = [];cw = []; ch = [];r = 0.75;for k=1:100    A = 4 + 1.*randn(sz);    A = A'*A;    W = diag(4 + 1.*randn(sz,1));    cw(1) = cond(W);    ch(1) = cond(W'*A*W);    [c, j] = min(diag(W));    for i=1:25        W(j,j) = W(j,j) * r;        cw(i+1) = cond(W);        ch(i+1) = cond(W'*A*W);    end    v(:,k) = (ch(2:end)./ch(1:end-1))';endplot(v);axis([1 i 1 1/r^2]);If I plot then ratios of condition numbers between iterations it approaches $1/r^2$. E.g., for $r = 0.75$:And it is never larger than $1/r^2$. It tells that perhaps there should be bound for condition number of the product $D^TAD$ or we can even compute it knowing spectrum of $A$, but I can't figure it out. Can anyone help?Also, this effect seems to be independent of random generator and its parameters. 
Spectrum of the product of two matrices
linear algebra
There is no useful relationship. You can see this by looking at the $2\times 2$ case, where explicit formulas exist.Edit; You can see from your plot that the worst cases in you sample curves are already very far from the typical cases. The absolutely worst would be even further off. Thus though one can perhaps prove optimal bounds (by writing down the optimatity conditions and analysing them), they would tell nothing useful about the typcal situation. 