# 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

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}- .

Because **I _{m}** is invertible, the formula for the determinant of a block matrix gives

- .

Because **I _{n}** 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.^{[5]}

## References

- ↑ 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. - ↑ Harville, David A. (2008).
*Matrix algebra from a statistician's perspective*. Berlin: Springer. ISBN 0-387-78356-3. page 416 - ↑ Weisstein, Eric W. "Sylvester's Determinant Identity". MathWorld--A Wolfram Web Resource. Retrieved 2012-03-03.
- ↑ Pozrikidis, C. (2014),
*An Introduction to Grids, Graphs, and Networks*, Oxford University Press, p. 271, ISBN 9780199996735 - ↑ "The mesoscopic structure of GUE eigenvalues | What's new".
*Terrytao.wordpress.com*. Retrieved 2016-01-16.