How to identify two matrices are congruent when they are not symmetric?
0
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It seems that most textbooks of Linear Algebra only mention Sylvester Theorem that tells us how to identify matrices are congruent when they are symmetric.
What happened if nonsymmetric? Is there any theorem or criterion that can be used to identify if two matrices are congruent when they are not symmetric?
linear-algebra matrices
asked Dec 22 '18 at 3:02
user450201user450201
1018
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add a comment |
0
$begingroup$
It seems that most textbooks of Linear Algebra only mention Sylvester Theorem that tells us how to identify matrices are congruent when they are symmetric.
What happened if nonsymmetric? Is there any theorem or criterion that can be used to identify if two matrices are congruent when they are not symmetric?
linear-algebra matrices
asked Dec 22 '18 at 3:02
user450201user450201
1018
$endgroup$
$begingroup$
Can you define congruency of matrices?
$endgroup$
– Aniruddh Agarwal
Dec 22 '18 at 3:43
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Two matrices $A$ and $B$ are congruent if there exist an invertible matrix $P$ such that $P^{T}AP=B$. @AniruddhAgarwal
$endgroup$
– user450201
Dec 22 '18 at 3:50
$begingroup$
Someone asked the same question recently on this site. For $T$-congruence of complex matrices, the answer is known (see the survey paper in the linked thread or Horn and Sergeichuk's papers mentioned in that survey paper), but I'm not sure about the real case, although in some special cases (e.g. when the symmetric parts of $A$ and $B$ are both positive definite, both negative definite or both zero) there are easy answers.
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– user1551
Dec 22 '18 at 6:37
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Thank you so much.@user1551
$endgroup$
– user450201
Dec 22 '18 at 6:44
$begingroup$
Where do we run into congruence? It's the change-of-basis relation for quadratic forms. The textbook theorem only deals with symmetric matrices because we only care about congruence for symmetric (or Hermitian, over $mathbb{C}$) matrices.
$endgroup$
– jmerry
Dec 22 '18 at 7:10
add a comment |
0
0
0
$begingroup$
It seems that most textbooks of Linear Algebra only mention Sylvester Theorem that tells us how to identify matrices are congruent when they are symmetric.
What happened if nonsymmetric? Is there any theorem or criterion that can be used to identify if two matrices are congruent when they are not symmetric?
linear-algebra matrices
asked Dec 22 '18 at 3:02
user450201user450201
1018
$endgroup$
It seems that most textbooks of Linear Algebra only mention Sylvester Theorem that tells us how to identify matrices are congruent when they are symmetric.
What happened if nonsymmetric? Is there any theorem or criterion that can be used to identify if two matrices are congruent when they are not symmetric?
linear-algebra matrices
linear-algebra matrices
asked Dec 22 '18 at 3:02
user450201user450201
1018
asked Dec 22 '18 at 3:02
user450201user450201
1018
asked Dec 22 '18 at 3:02
user450201user450201
1018
asked Dec 22 '18 at 3:02
user450201user450201
1018
asked Dec 22 '18 at 3:02
user450201user450201
1018
1018
$begingroup$
Can you define congruency of matrices?
$endgroup$
– Aniruddh Agarwal
Dec 22 '18 at 3:43
$begingroup$
Two matrices $A$ and $B$ are congruent if there exist an invertible matrix $P$ such that $P^{T}AP=B$. @AniruddhAgarwal
$endgroup$
– user450201
Dec 22 '18 at 3:50
$begingroup$
Someone asked the same question recently on this site. For $T$-congruence of complex matrices, the answer is known (see the survey paper in the linked thread or Horn and Sergeichuk's papers mentioned in that survey paper), but I'm not sure about the real case, although in some special cases (e.g. when the symmetric parts of $A$ and $B$ are both positive definite, both negative definite or both zero) there are easy answers.
$endgroup$
– user1551
Dec 22 '18 at 6:37
$begingroup$
Thank you so much.@user1551
$endgroup$
– user450201
Dec 22 '18 at 6:44
$begingroup$
Where do we run into congruence? It's the change-of-basis relation for quadratic forms. The textbook theorem only deals with symmetric matrices because we only care about congruence for symmetric (or Hermitian, over $mathbb{C}$) matrices.
$endgroup$
– jmerry
Dec 22 '18 at 7:10
add a comment |
$begingroup$
Can you define congruency of matrices?
$endgroup$
– Aniruddh Agarwal
Dec 22 '18 at 3:43
$begingroup$
Two matrices $A$ and $B$ are congruent if there exist an invertible matrix $P$ such that $P^{T}AP=B$. @AniruddhAgarwal
$endgroup$
– user450201
Dec 22 '18 at 3:50
$begingroup$
Someone asked the same question recently on this site. For $T$-congruence of complex matrices, the answer is known (see the survey paper in the linked thread or Horn and Sergeichuk's papers mentioned in that survey paper), but I'm not sure about the real case, although in some special cases (e.g. when the symmetric parts of $A$ and $B$ are both positive definite, both negative definite or both zero) there are easy answers.
$endgroup$
– user1551
Dec 22 '18 at 6:37
$begingroup$
Thank you so much.@user1551
$endgroup$
– user450201
Dec 22 '18 at 6:44
$begingroup$
Where do we run into congruence? It's the change-of-basis relation for quadratic forms. The textbook theorem only deals with symmetric matrices because we only care about congruence for symmetric (or Hermitian, over $mathbb{C}$) matrices.
$endgroup$
– jmerry
Dec 22 '18 at 7:10
$begingroup$
Can you define congruency of matrices?
$endgroup$
– Aniruddh Agarwal
Dec 22 '18 at 3:43
$begingroup$
Can you define congruency of matrices?
$endgroup$
– Aniruddh Agarwal
Dec 22 '18 at 3:43
$begingroup$
Two matrices $A$ and $B$ are congruent if there exist an invertible matrix $P$ such that $P^{T}AP=B$. @AniruddhAgarwal
$endgroup$
– user450201
Dec 22 '18 at 3:50
$begingroup$
Two matrices $A$ and $B$ are congruent if there exist an invertible matrix $P$ such that $P^{T}AP=B$. @AniruddhAgarwal
$endgroup$
– user450201
Dec 22 '18 at 3:50
$begingroup$
Someone asked the same question recently on this site. For $T$-congruence of complex matrices, the answer is known (see the survey paper in the linked thread or Horn and Sergeichuk's papers mentioned in that survey paper), but I'm not sure about the real case, although in some special cases (e.g. when the symmetric parts of $A$ and $B$ are both positive definite, both negative definite or both zero) there are easy answers.
$endgroup$
– user1551
Dec 22 '18 at 6:37
$begingroup$
Someone asked the same question recently on this site. For $T$-congruence of complex matrices, the answer is known (see the survey paper in the linked thread or Horn and Sergeichuk's papers mentioned in that survey paper), but I'm not sure about the real case, although in some special cases (e.g. when the symmetric parts of $A$ and $B$ are both positive definite, both negative definite or both zero) there are easy answers.
$endgroup$
– user1551
Dec 22 '18 at 6:37
$begingroup$
Thank you so much.@user1551
$endgroup$
– user450201
Dec 22 '18 at 6:44
$begingroup$
Thank you so much.@user1551
$endgroup$
– user450201
Dec 22 '18 at 6:44
$begingroup$
Where do we run into congruence? It's the change-of-basis relation for quadratic forms. The textbook theorem only deals with symmetric matrices because we only care about congruence for symmetric (or Hermitian, over $mathbb{C}$) matrices.
$endgroup$
– jmerry
Dec 22 '18 at 7:10
$begingroup$
Where do we run into congruence? It's the change-of-basis relation for quadratic forms. The textbook theorem only deals with symmetric matrices because we only care about congruence for symmetric (or Hermitian, over $mathbb{C}$) matrices.
$endgroup$
– jmerry
Dec 22 '18 at 7:10
add a comment |
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$begingroup$
Can you define congruency of matrices?
$endgroup$
– Aniruddh Agarwal
Dec 22 '18 at 3:43
$begingroup$
Two matrices $A$ and $B$ are congruent if there exist an invertible matrix $P$ such that $P^{T}AP=B$. @AniruddhAgarwal
$endgroup$
– user450201
Dec 22 '18 at 3:50
$begingroup$
Someone asked the same question recently on this site. For $T$-congruence of complex matrices, the answer is known (see the survey paper in the linked thread or Horn and Sergeichuk's papers mentioned in that survey paper), but I'm not sure about the real case, although in some special cases (e.g. when the symmetric parts of $A$ and $B$ are both positive definite, both negative definite or both zero) there are easy answers.
$endgroup$
– user1551
Dec 22 '18 at 6:37
$begingroup$
Thank you so much.@user1551
$endgroup$
– user450201
Dec 22 '18 at 6:44
$begingroup$
Where do we run into congruence? It's the change-of-basis relation for quadratic forms. The textbook theorem only deals with symmetric matrices because we only care about congruence for symmetric (or Hermitian, over $mathbb{C}$) matrices.
$endgroup$
– jmerry
Dec 22 '18 at 7:10