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Question:

prove or disprove

for any matrix $A$ and $A^T$ are Matrix congruence?

This problem is my Suddenly thought today .

I know this fact:

if $A$ is symmetry matrix,then $A$ and $A^T$ are matrix congruence.

Now this problem not tell us $A$ is symmetry.

so How prove it? Thank you very much

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    $A$ and $A^t$ are similar: http://math.stackexchange.com/questions/62497/matrix-is-conjugate-to-its-own-transpose – martini Dec 11 '13 at 14:13
  • No,But even $A$ and $A^T$ are similar,we can't have $A$ and $A^T$ are Matrix congruence. –  Dec 11 '13 at 14:20
  • What do you mean by "are matrix congruence"? By definition $A$ and $B$ are similar if there exists invertible $P$ with $PAP^{-1}=B$, is that not what you mean? – Marc van Leeuwen Dec 11 '13 at 14:36
  • The fact that if $A$ is symmetric then $A$ and $A^T$ "are matrix congruence" does not help us much, since they are the same. – Marc van Leeuwen Dec 11 '13 at 14:37
  • I think the OP meant to ask whether $A$ and $A^T$ are always congruent to each other, i.e. there exists a nonsignular matrix $C$ such that $C^TAC=A^T$. He/she is not talking about similarity and we don't require that $C^T=C^{-1}$. – user1551 Dec 11 '13 at 16:56

1 Answers1

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The answer is affirmative over any field. See

Dragomir Ž. Đoković, Khakim D. Ikramov (2002), A square matrix is congruent to its transpose, Journal of Algebra, 257(1): 97-105.

When the field is complex, not only is $A$ congruent to $A^T$, it is $\ast$-congruent to $A^T$ as well. See

Roger A. Horn, Vladimir V. Sergeichuk (2004), Congruences of a square matrix and its transpose, Linear Algebra and its Applications, 389: 347-353. (Also available from arxiv.org .)

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