Eigen Values of a product of two matrices
Suppose we have a matrix $C = DP$ where D is a real diagonal matrix with all the entries positive, and P is a real matrix whose eigen values are positive and all the elements are positive. Moreover, each row of P sums to the same value. Can we conclude that C has all the eigen values positive?
linear-algebra
asked Nov 28 at 13:07
Suhan Shetty
1077
add a comment |
Suppose we have a matrix $C = DP$ where D is a real diagonal matrix with all the entries positive, and P is a real matrix whose eigen values are positive and all the elements are positive. Moreover, each row of P sums to the same value. Can we conclude that C has all the eigen values positive?
linear-algebra
asked Nov 28 at 13:07
Suhan Shetty
1077
If the matrix $P$ is symmetric (in addition to the above conditions), I suspect that $C$ can be shown to have all positive eigenvalues. But I'm not 100% sure.
– Michael Seifert
Nov 28 at 14:01
@MichaelSeifert Indeed. If $P$ is symmetric, then $C=DP$ is similar to $D^{-1/2}CD^{1/2}=D^{1/2}PD^{1/2}$, which is positive definite.
– user1551
Nov 29 at 11:41
Why is $D^{1/2}PD{1/2}$ positive definite if P is symmetric? @user1551
– Suhan Shetty
Nov 29 at 12:26
@SuhanShetty In your question, $P$ is assumed to be a real matrix with a positive spectrum. If it is also symmetric, it is positive definite. Hence $D^{1/2}PD^{1/2}$ (which is congruent to $P$) is positive definite too.
– user1551
Nov 29 at 12:42
add a comment |
Suppose we have a matrix $C = DP$ where D is a real diagonal matrix with all the entries positive, and P is a real matrix whose eigen values are positive and all the elements are positive. Moreover, each row of P sums to the same value. Can we conclude that C has all the eigen values positive?
linear-algebra
asked Nov 28 at 13:07
Suhan Shetty
1077
Suppose we have a matrix $C = DP$ where D is a real diagonal matrix with all the entries positive, and P is a real matrix whose eigen values are positive and all the elements are positive. Moreover, each row of P sums to the same value. Can we conclude that C has all the eigen values positive?
linear-algebra
linear-algebra
asked Nov 28 at 13:07
Suhan Shetty
1077
asked Nov 28 at 13:07
Suhan Shetty
1077
asked Nov 28 at 13:07
Suhan Shetty
1077
asked Nov 28 at 13:07
Suhan Shetty
1077
asked Nov 28 at 13:07
Suhan Shetty
1077
1077
If the matrix $P$ is symmetric (in addition to the above conditions), I suspect that $C$ can be shown to have all positive eigenvalues. But I'm not 100% sure.
– Michael Seifert
Nov 28 at 14:01
@MichaelSeifert Indeed. If $P$ is symmetric, then $C=DP$ is similar to $D^{-1/2}CD^{1/2}=D^{1/2}PD^{1/2}$, which is positive definite.
– user1551
Nov 29 at 11:41
Why is $D^{1/2}PD{1/2}$ positive definite if P is symmetric? @user1551
– Suhan Shetty
Nov 29 at 12:26
@SuhanShetty In your question, $P$ is assumed to be a real matrix with a positive spectrum. If it is also symmetric, it is positive definite. Hence $D^{1/2}PD^{1/2}$ (which is congruent to $P$) is positive definite too.
– user1551
Nov 29 at 12:42
add a comment |
If the matrix $P$ is symmetric (in addition to the above conditions), I suspect that $C$ can be shown to have all positive eigenvalues. But I'm not 100% sure.
– Michael Seifert
Nov 28 at 14:01
@MichaelSeifert Indeed. If $P$ is symmetric, then $C=DP$ is similar to $D^{-1/2}CD^{1/2}=D^{1/2}PD^{1/2}$, which is positive definite.
– user1551
Nov 29 at 11:41
Why is $D^{1/2}PD{1/2}$ positive definite if P is symmetric? @user1551
– Suhan Shetty
Nov 29 at 12:26
@SuhanShetty In your question, $P$ is assumed to be a real matrix with a positive spectrum. If it is also symmetric, it is positive definite. Hence $D^{1/2}PD^{1/2}$ (which is congruent to $P$) is positive definite too.
– user1551
Nov 29 at 12:42
If the matrix $P$ is symmetric (in addition to the above conditions), I suspect that $C$ can be shown to have all positive eigenvalues. But I'm not 100% sure.
– Michael Seifert
Nov 28 at 14:01
If the matrix $P$ is symmetric (in addition to the above conditions), I suspect that $C$ can be shown to have all positive eigenvalues. But I'm not 100% sure.
– Michael Seifert
Nov 28 at 14:01
@MichaelSeifert Indeed. If $P$ is symmetric, then $C=DP$ is similar to $D^{-1/2}CD^{1/2}=D^{1/2}PD^{1/2}$, which is positive definite.
– user1551
Nov 29 at 11:41
@MichaelSeifert Indeed. If $P$ is symmetric, then $C=DP$ is similar to $D^{-1/2}CD^{1/2}=D^{1/2}PD^{1/2}$, which is positive definite.
– user1551
Nov 29 at 11:41
Why is $D^{1/2}PD{1/2}$ positive definite if P is symmetric? @user1551
– Suhan Shetty
Nov 29 at 12:26
Why is $D^{1/2}PD{1/2}$ positive definite if P is symmetric? @user1551
– Suhan Shetty
Nov 29 at 12:26
@SuhanShetty In your question, $P$ is assumed to be a real matrix with a positive spectrum. If it is also symmetric, it is positive definite. Hence $D^{1/2}PD^{1/2}$ (which is congruent to $P$) is positive definite too.
– user1551
Nov 29 at 12:42
@SuhanShetty In your question, $P$ is assumed to be a real matrix with a positive spectrum. If it is also symmetric, it is positive definite. Hence $D^{1/2}PD^{1/2}$ (which is congruent to $P$) is positive definite too.
– user1551
Nov 29 at 12:42
add a comment |
1 Answer
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No. Random counterexample:
$$
P=pmatrix{3&2&5\ 4&5&1\ 3&3&4}, D=operatorname{diag}(8,1,7).
$$
The eigenvalues of $P$ are $10,1,1$ but the eigenvalues of $DP$ are $57.05$ and $-0.025pm3.13i$.
answered Nov 28 at 13:58
user1551
71.2k566125
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
No. Random counterexample:
$$
P=pmatrix{3&2&5\ 4&5&1\ 3&3&4}, D=operatorname{diag}(8,1,7).
$$
The eigenvalues of $P$ are $10,1,1$ but the eigenvalues of $DP$ are $57.05$ and $-0.025pm3.13i$.
answered Nov 28 at 13:58
user1551
71.2k566125
add a comment |
No. Random counterexample:
$$
P=pmatrix{3&2&5\ 4&5&1\ 3&3&4}, D=operatorname{diag}(8,1,7).
$$
The eigenvalues of $P$ are $10,1,1$ but the eigenvalues of $DP$ are $57.05$ and $-0.025pm3.13i$.
answered Nov 28 at 13:58
user1551
71.2k566125
add a comment |
No. Random counterexample:
$$
P=pmatrix{3&2&5\ 4&5&1\ 3&3&4}, D=operatorname{diag}(8,1,7).
$$
The eigenvalues of $P$ are $10,1,1$ but the eigenvalues of $DP$ are $57.05$ and $-0.025pm3.13i$.
answered Nov 28 at 13:58
user1551
71.2k566125
No. Random counterexample:
$$
P=pmatrix{3&2&5\ 4&5&1\ 3&3&4}, D=operatorname{diag}(8,1,7).
$$
The eigenvalues of $P$ are $10,1,1$ but the eigenvalues of $DP$ are $57.05$ and $-0.025pm3.13i$.
answered Nov 28 at 13:58
user1551
71.2k566125
answered Nov 28 at 13:58
user1551
71.2k566125
answered Nov 28 at 13:58
user1551
71.2k566125
answered Nov 28 at 13:58
user1551
71.2k566125
71.2k566125
add a comment |
add a comment |
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If the matrix $P$ is symmetric (in addition to the above conditions), I suspect that $C$ can be shown to have all positive eigenvalues. But I'm not 100% sure.
– Michael Seifert
Nov 28 at 14:01
@MichaelSeifert Indeed. If $P$ is symmetric, then $C=DP$ is similar to $D^{-1/2}CD^{1/2}=D^{1/2}PD^{1/2}$, which is positive definite.
– user1551
Nov 29 at 11:41
Why is $D^{1/2}PD{1/2}$ positive definite if P is symmetric? @user1551
– Suhan Shetty
Nov 29 at 12:26
@SuhanShetty In your question, $P$ is assumed to be a real matrix with a positive spectrum. If it is also symmetric, it is positive definite. Hence $D^{1/2}PD^{1/2}$ (which is congruent to $P$) is positive definite too.
– user1551
Nov 29 at 12:42