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))';
end
plot(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.