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

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.

Since $A^{-1}=(A^{-1})^T$, it follows that $P^TAP=A^{-1}$ holds for $P=A^{-1}$.
• Means you are taking P particularly as $A^{-1}$ – rama_ran Sep 5 '16 at 5:06
• Insted of proving that there exists $P$ such that $P^TAP=A^{-1}$ holds, we give the explicit expression for $P$. – Ignat Domanov Sep 5 '16 at 5:20
• @Ignat "Instead of proving that there exists $P$ such that $P^TAP=A^{-1}$ holds..." It is exactly what you proved. – Algebraic Pavel Sep 5 '16 at 9:37