Sylvester's matrix theorem

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In Matrix theory, Sylvester's matrix theorem allows one to evaluate functions of matrices easily.

Suppose A is a square matrix of size . If the eigenvalues of A are λ_i for , and the corresponding normalized row and column eigenvectors are r_i and c_i respectively, then the theorem states that

[edit] Example

Consider a two-by-two matrix:

and a function f() that squares its argument: f(A) = AA = A².

Matrix A has eigenvalues 5 and -2. The row eigenvectors are (1 / 7,1 / 7) and (4, − 3); the column eigenvectors are (3,4)^T and (1 / 7, − 1 / 7)^T; the 7 is a normalization factor.

Thus and .

Sylvester's matrix theorem states that

as required.

Sylvester's theorem is useful for calculating computationally demanding functions such as matrix exponentials e^A.

[edit] References