Sylvester's determinant identity

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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 Ia 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[edit]

The identity may be proved as follows.[4] Let M be a matrix comprising the four blocks Im, A, B, and In:

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

Because Im 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 In 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[edit]

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[edit]

  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.