Mixing
Change of eigenvalues of a matrix when pre- and post multiplied by a diagonal matrix
0
$begingroup$
Let $Ain mathbb{R}^{ntimes n}$. Moreover, assume $D$ is an $n times n$ diagonal matrix with positive diagonals. What is the relation between the eigenvalues of $A$ and eigenvalues of $B:=DAD$? In other words, how do the eigenvalues of the matrix $A$ change when it is pre and post multiplied by a diagonal matrix? Do $A$ and $B$ have the same inertia?
We can assume that $A$ is diagonalizable if necessary.
Any comment/response is greatly appreciated.
linear-algebra matrices eigenvalues-eigenvectors
asked Jan 12 at 20:34
ArthurArthur
49512
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show 4 more comments
0
$begingroup$
Let $Ain mathbb{R}^{ntimes n}$. Moreover, assume $D$ is an $n times n$ diagonal matrix with positive diagonals. What is the relation between the eigenvalues of $A$ and eigenvalues of $B:=DAD$? In other words, how do the eigenvalues of the matrix $A$ change when it is pre and post multiplied by a diagonal matrix? Do $A$ and $B$ have the same inertia?
We can assume that $A$ is diagonalizable if necessary.
Any comment/response is greatly appreciated.
linear-algebra matrices eigenvalues-eigenvectors
asked Jan 12 at 20:34
ArthurArthur
49512
$endgroup$
$begingroup$
$A$ and $B$ are congruent (since $D=D^T$) so they do have the same inertia
$endgroup$
– Omnomnomnom
Jan 12 at 20:49
$begingroup$
Thanks for the comment. How about the location of eigenvalues. Is there any result on how the eigenvalues of $A$ would move?
$endgroup$
– Arthur
Jan 12 at 20:55
$begingroup$
Is $A$ symmetric?
$endgroup$
– Omnomnomnom
Jan 12 at 21:09
1
$begingroup$
Bhatia's Matrix Analysis is usually a good reference for these purposes, but I'm having trouble finding inequalities that would apply to your specific situation. If you were content to consider the singular values rather than eigenvalues, then we could note that $DAD$ is similar to $D^2 A$, and that $$ prod_{j=1}^k sigma_j(D^2A) leq prod_{j=1}^k sigma_j(D^2) prod_{j=1}^k sigma_j(A) $$ The singular values of $D^2$, since $D^2$ is diagonal with positive diagonals, are just those values on the diagonal.
$endgroup$
– Omnomnomnom
Jan 13 at 16:51
1
$begingroup$
You can find more inequalities along these lines in section III.4 of Bhatia's text (Lidskii's theorems)
$endgroup$
– Omnomnomnom
Jan 13 at 16:53
|
show 4 more comments
0
0
0
$begingroup$
Let $Ain mathbb{R}^{ntimes n}$. Moreover, assume $D$ is an $n times n$ diagonal matrix with positive diagonals. What is the relation between the eigenvalues of $A$ and eigenvalues of $B:=DAD$? In other words, how do the eigenvalues of the matrix $A$ change when it is pre and post multiplied by a diagonal matrix? Do $A$ and $B$ have the same inertia?
We can assume that $A$ is diagonalizable if necessary.
Any comment/response is greatly appreciated.
linear-algebra matrices eigenvalues-eigenvectors
asked Jan 12 at 20:34
ArthurArthur
49512
$endgroup$
Let $Ain mathbb{R}^{ntimes n}$. Moreover, assume $D$ is an $n times n$ diagonal matrix with positive diagonals. What is the relation between the eigenvalues of $A$ and eigenvalues of $B:=DAD$? In other words, how do the eigenvalues of the matrix $A$ change when it is pre and post multiplied by a diagonal matrix? Do $A$ and $B$ have the same inertia?
We can assume that $A$ is diagonalizable if necessary.
Any comment/response is greatly appreciated.
linear-algebra matrices eigenvalues-eigenvectors
linear-algebra matrices eigenvalues-eigenvectors
asked Jan 12 at 20:34
ArthurArthur
49512
asked Jan 12 at 20:34
ArthurArthur
49512
asked Jan 12 at 20:34
ArthurArthur
49512
asked Jan 12 at 20:34
ArthurArthur
49512
asked Jan 12 at 20:34
ArthurArthur
49512
49512
$begingroup$
$A$ and $B$ are congruent (since $D=D^T$) so they do have the same inertia
$endgroup$
– Omnomnomnom
Jan 12 at 20:49
$begingroup$
Thanks for the comment. How about the location of eigenvalues. Is there any result on how the eigenvalues of $A$ would move?
$endgroup$
– Arthur
Jan 12 at 20:55
$begingroup$
Is $A$ symmetric?
$endgroup$
– Omnomnomnom
Jan 12 at 21:09
1
$begingroup$
Bhatia's Matrix Analysis is usually a good reference for these purposes, but I'm having trouble finding inequalities that would apply to your specific situation. If you were content to consider the singular values rather than eigenvalues, then we could note that $DAD$ is similar to $D^2 A$, and that $$ prod_{j=1}^k sigma_j(D^2A) leq prod_{j=1}^k sigma_j(D^2) prod_{j=1}^k sigma_j(A) $$ The singular values of $D^2$, since $D^2$ is diagonal with positive diagonals, are just those values on the diagonal.
$endgroup$
– Omnomnomnom
Jan 13 at 16:51
1
$begingroup$
You can find more inequalities along these lines in section III.4 of Bhatia's text (Lidskii's theorems)
$endgroup$
– Omnomnomnom
Jan 13 at 16:53
|
show 4 more comments
$begingroup$
$A$ and $B$ are congruent (since $D=D^T$) so they do have the same inertia
$endgroup$
– Omnomnomnom
Jan 12 at 20:49
$begingroup$
Thanks for the comment. How about the location of eigenvalues. Is there any result on how the eigenvalues of $A$ would move?
$endgroup$
– Arthur
Jan 12 at 20:55
$begingroup$
Is $A$ symmetric?
$endgroup$
– Omnomnomnom
Jan 12 at 21:09
1
$begingroup$
Bhatia's Matrix Analysis is usually a good reference for these purposes, but I'm having trouble finding inequalities that would apply to your specific situation. If you were content to consider the singular values rather than eigenvalues, then we could note that $DAD$ is similar to $D^2 A$, and that $$ prod_{j=1}^k sigma_j(D^2A) leq prod_{j=1}^k sigma_j(D^2) prod_{j=1}^k sigma_j(A) $$ The singular values of $D^2$, since $D^2$ is diagonal with positive diagonals, are just those values on the diagonal.
$endgroup$
– Omnomnomnom
Jan 13 at 16:51
1
$begingroup$
You can find more inequalities along these lines in section III.4 of Bhatia's text (Lidskii's theorems)
$endgroup$
– Omnomnomnom
Jan 13 at 16:53
$begingroup$
$A$ and $B$ are congruent (since $D=D^T$) so they do have the same inertia
$endgroup$
– Omnomnomnom
Jan 12 at 20:49
$begingroup$
$A$ and $B$ are congruent (since $D=D^T$) so they do have the same inertia
$endgroup$
– Omnomnomnom
Jan 12 at 20:49
$begingroup$
Thanks for the comment. How about the location of eigenvalues. Is there any result on how the eigenvalues of $A$ would move?
$endgroup$
– Arthur
Jan 12 at 20:55
$begingroup$
Thanks for the comment. How about the location of eigenvalues. Is there any result on how the eigenvalues of $A$ would move?
$endgroup$
– Arthur
Jan 12 at 20:55
$begingroup$
Is $A$ symmetric?
$endgroup$
– Omnomnomnom
Jan 12 at 21:09
$begingroup$
Is $A$ symmetric?
$endgroup$
– Omnomnomnom
Jan 12 at 21:09
1
1
$begingroup$
Bhatia's Matrix Analysis is usually a good reference for these purposes, but I'm having trouble finding inequalities that would apply to your specific situation. If you were content to consider the singular values rather than eigenvalues, then we could note that $DAD$ is similar to $D^2 A$, and that $$ prod_{j=1}^k sigma_j(D^2A) leq prod_{j=1}^k sigma_j(D^2) prod_{j=1}^k sigma_j(A) $$ The singular values of $D^2$, since $D^2$ is diagonal with positive diagonals, are just those values on the diagonal.
$endgroup$
– Omnomnomnom
Jan 13 at 16:51
$begingroup$
Bhatia's Matrix Analysis is usually a good reference for these purposes, but I'm having trouble finding inequalities that would apply to your specific situation. If you were content to consider the singular values rather than eigenvalues, then we could note that $DAD$ is similar to $D^2 A$, and that $$ prod_{j=1}^k sigma_j(D^2A) leq prod_{j=1}^k sigma_j(D^2) prod_{j=1}^k sigma_j(A) $$ The singular values of $D^2$, since $D^2$ is diagonal with positive diagonals, are just those values on the diagonal.
$endgroup$
– Omnomnomnom
Jan 13 at 16:51
1
1
$begingroup$
You can find more inequalities along these lines in section III.4 of Bhatia's text (Lidskii's theorems)
$endgroup$
– Omnomnomnom
Jan 13 at 16:53
$begingroup$
You can find more inequalities along these lines in section III.4 of Bhatia's text (Lidskii's theorems)
$endgroup$
– Omnomnomnom
Jan 13 at 16:53
|
show 4 more comments
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$begingroup$
$A$ and $B$ are congruent (since $D=D^T$) so they do have the same inertia
$endgroup$
– Omnomnomnom
Jan 12 at 20:49
$begingroup$
Thanks for the comment. How about the location of eigenvalues. Is there any result on how the eigenvalues of $A$ would move?
$endgroup$
– Arthur
Jan 12 at 20:55
$begingroup$
Is $A$ symmetric?
$endgroup$
– Omnomnomnom
Jan 12 at 21:09
1
$begingroup$
Bhatia's Matrix Analysis is usually a good reference for these purposes, but I'm having trouble finding inequalities that would apply to your specific situation. If you were content to consider the singular values rather than eigenvalues, then we could note that $DAD$ is similar to $D^2 A$, and that $$ prod_{j=1}^k sigma_j(D^2A) leq prod_{j=1}^k sigma_j(D^2) prod_{j=1}^k sigma_j(A) $$ The singular values of $D^2$, since $D^2$ is diagonal with positive diagonals, are just those values on the diagonal.
$endgroup$
– Omnomnomnom
Jan 13 at 16:51
1
$begingroup$
You can find more inequalities along these lines in section III.4 of Bhatia's text (Lidskii's theorems)
$endgroup$
– Omnomnomnom
Jan 13 at 16:53