Sylvester's determinant identity

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.^[1]

The identity states that if A and B are matrices of size m × n and n × m respectively, then

$\det(I_m + AB) = \det(I_n + BA),\$

where I_a is the identity matrix of order a.^[2][3]

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.^[4] Let M be a matrix comprising the four blocks I_m, −A, B, and I_n:

$M = \begin{pmatrix}I_m & -A \\ B & I_n \end{pmatrix}$.

Because I_m 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 I_n 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

$\det(I_n + B A) = \det(I_m + A B)$.

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.^[5]

References

1. Sylvester, James Joseph (1851). "On the relation between the minor determinants of linearly equivalent quadratic functions". Philosophical Magazine 1: 295–305.
   Cited in Akritas, A. G.; Akritas, E. K.; Malaschonok, G. I. (1996). "Various proofs of Sylvester's (determinant) identity". Mathematics and Computers in Simulation 42 (4–6): 585. doi:10.1016/S0378-4754(96)00035-3.
2. Harville, David A. (2008). Matrix algebra from a statistician's perspective. Berlin: Springer. ISBN 0-387-78356-3. page 416
3. Weisstein, Eric W. "Sylvester's Determinant Identity". MathWorld--A Wolfram Web Resource. Retrieved 2012-03-03.
4. Pozrikidis, C. (2014), An Introduction to Grids, Graphs, and Networks, Oxford University Press, p. 271, ISBN 9780199996735
5. "The mesoscopic structure of GUE eigenvalues | What's new". Terrytao.wordpress.com. Retrieved 2016-01-16.