Note: this page is better understood with knowledge of symmetric positive-definite matrices and quadratic forms.
A congruent transformation on a square matrix \(A \in \mathbb{R}^{m \times m}\) has the form $$ A' = W^T A W $$ where \(W\) is a (possibly non-square) matrix \(W \in \mathbb{R}^{m \times n}\). While this transformation bears some surface-level resemblance to a similarity transform with the inverse replaced with a transpose, we note that this resemblance is actually somewhat deceptive. Assuming \(A\) is symmetric, congruent transformations preserve symmetries and definiteness of the matrix (and thus the signs of the eigenvalues) but not the eigenvalues of the matrix. The one exception is when \(W\) is a rotation and then the congruent transformation is also a similarity transformation.
Symmetric \(A\)In the case when \(A\) is symmetric, a congruent transformation gives how the quadratic form \(x^TAx\) changes under the coordinate transformation \(x = Wx'\). Explicitly, $$ x^TAx = x'^TW^TAWx' = x'^TA'x' $$ Below we give a visualization of the level sets of \(x^TAx\) and how the the transform under \(x = Wx'\).
