Matrix congruence

Remarks

Example: over $\R$, let $A = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ and $P = \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}$. Then $P$ is invertible, since $\det P = -2$, and $P^{T} A P = \begin{pmatrix} 2 & 0 \\ 0 & -2 \end{pmatrix}$, so these two matrices are congruent. Non-example: over $\R$, the identity $I_2$ and $-I_2$ are not congruent, because $x^{T} P^{T} I_2 P x = \norm{Px}^2 \ge 0$ for all $x \in \R^2$, while $x^{T} (-I_2) x < 0$ for $x \ne 0$. Congruence is an equivalence relation on $n \times n$ matrices. Let $A$ be the matrix of a bilinear form in one basis, and let $P$ be the change of basis matrix to a second basis. Then $P^{T} A P$ is the matrix of the same form in the second basis; see change of basis for a bilinear form. Congruence keeps symmetry and the rank, and it multiplies the determinant by the nonzero square $(\det P)^2$. Over $\C$, the complex analogue often uses the conjugate transpose $P^{*}$ in place of $P^{T}$; this is called $*$-congruence.