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Show that every nonsingular symmetric matrix is congruent to its inverse.

I know that Two matrices A and B over a field are called congruent if there exists an invertible matrix P over the same field such that $$P^TAP = B$$

But how to prove the above.

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Since $A^{-1}=(A^{-1})^T$, it follows that $P^TAP=A^{-1}$ holds for $P=A^{-1}$.

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  • $\begingroup$ This is a particular case. Is it not? I need a generalized proof. $\endgroup$ – rama_ran Sep 5 '16 at 4:38
  • $\begingroup$ What do you mean by "particular case"?clarify please. $\endgroup$ – Ignat Domanov Sep 5 '16 at 4:46
  • $\begingroup$ Means you are taking P particularly as $A^{-1}$ $\endgroup$ – rama_ran Sep 5 '16 at 5:06
  • $\begingroup$ Insted of proving that there exists $P$ such that $P^TAP=A^{-1}$ holds, we give the explicit expression for $P$. $\endgroup$ – Ignat Domanov Sep 5 '16 at 5:20
  • $\begingroup$ @Ignat "Instead of proving that there exists $P$ such that $P^TAP=A^{-1}$ holds..." It is exactly what you proved. $\endgroup$ – Algebraic Pavel Sep 5 '16 at 9:37

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