Sylvester's determinant theorem
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In matrix theory, Sylvester's determinant theorem is a theorem useful for evaluating certain types of determinants. It is named for James Joseph Sylvester.
The theorem states that if A,B are matrices of size 
and 
respectively, then
- det(Ip + AB) = det(In + BA),
where Ia denotes the identity matrix of order a.
This theorem is useful in developing a Bayes estimator for multivariate Gaussian distributions.
Sylvester (1857) stated this theorem without proof.
Despite the fact that the importance of Sylvester's determinant identity has been recognized in the past, only one proof of it in English (Bareiss, 1968) is widely known. Other proofs exist (four in German and two in Russian).
[edit] Reference
- Akritas A.G.1; Akritas E.K.; Malaschonok G.I., "Various proofs of Sylvester's (determinant) identity", Mathematics and Computers in Simulation, Volume 42, Number 4, November 1996, pp. 585-593(9), Elsevier
