In matrix theory, Sylvester's determinant identity is an identity useful for evaluating certain types of determinants. It is named after James Joseph Sylvester, who stated this identity without proof in 1851.
Contents
The identity states that if A and B are matrices of size m × n and n × m respectively, then
where Ia is the identity matrix of order a.
It is closely related to the Matrix determinant lemma and its generalization. It is the determinant analogue of the Woodbury matrix identity for matrix inverses.
Proof
The identity may be proved as follows. Let M be a matrix comprising the four blocks Im, −A, B, and In:
Because Im is invertible, the formula for the determinant of a block matrix gives
Because In is invertible, the formula for the determinant of a block matrix gives
Thus
Applications
This identify 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.