Weinstein–Aronszajn identity

The identity may be proved as follows. Let M be a matrix consisting of the four blocks I_m, A, B and I_n: :M = \begin{pmatrix} I_m & A \\ B & I_n \end{pmatrix}. Because is invertible, the formula for the determinant of a block matrix gives :\det\!\begin{pmatrix} I_m & A \\ B & I_n \end{pmatrix} = \det(I_m) \det(I_n - B I_m^{-1} A) = \det(I_n - BA). Because is invertible, the formula for the determinant of a block matrix gives :\det\!\begin{pmatrix} I_m & A\\ B & I_n \end{pmatrix} = \det(I_n) \det(I_m - A I_n^{-1} B) = \det(I_m - AB). Thus :\det(I_n - B A) = \det(I_m - A B). Substituting -A for A then gives the Weinstein–Aronszajn identity. ==Applications==

Let \lambda \in \mathbb{R} \setminus \{0\}. The identity can be used to show the somewhat more general statement that : \det(AB - \lambda I_m) = (-\lambda)^{m - n} \det(BA - \lambda I_n). It follows that the non-zero eigenvalues of AB and BA are the same. This identity is useful in developing a Bayes estimator for multivariate Gaussian distributions. The identity also finds applications in random matrix theory by relating determinants of large matrices to determinants of smaller ones. ==References==