In matrix theory, Sylvester's formula or Sylvester's matrix theorem (named after J. J. Sylvester

James Joseph Sylvester

James Joseph Sylvester was an English mathematician. He made fundamental contributions to matrix theory, invariant theory, number theory, partition theory and combinatorics...

) expresses an analytic function

Matrix function

In mathematics, a matrix function is a function which maps a matrix to another matrix.- Extending scalar functions to matrix functions :There are several techniques for lifting a real function to a square matrix function such that interesting properties are maintained...

 f(A) of a matrix

Matrix (mathematics)

In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...

 A in terms of the eigenvalues and eigenvectors

Eigenvalue, eigenvector and eigenspace

The eigenvectors of a square matrix are the non-zero vectors that, after being multiplied by the matrix, remain parallel to the original vector. For each eigenvector, the corresponding eigenvalue is the factor by which the eigenvector is scaled when multiplied by the matrix...

 of A. It states that

where the λ_i are the eigenvalues of A, and the matrices A_i are the corresponding Frobenius covariant

Frobenius covariant

In matrix theory, the Frobenius covariants of a square matrix A are matrices Ai associated with the eigenvalues and eigenvectors of A...

s of A.

Sylvester's formula is only valid for diagonalizable matrices

Diagonalizable matrix

In linear algebra, a square matrix A is called diagonalizable if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix P such that P −1AP is a diagonal matrix...

; an extension due to Buchheim covers the general case.

Conditions

Sylvester's formula applies for any diagonalizable matrix

Diagonalizable matrix

In linear algebra, a square matrix A is called diagonalizable if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix P such that P −1AP is a diagonal matrix...

 A with k distinct eigenvalues, λ₁, …, λ_k, and any function f defined on some subset of the complex numbers such that f(A) is well defined. The last condition means that every eigenvalue λ_i is in the domain of f, and that every eigenvalue λ_i with multiplicity m_i > 1 is in the interior of the domain, with f being (m_i − 1) times differentiable at λ_i..

Example

Consider the two-by-two matrix:


This matrix has two eigenvalues, 5 and −2. Its Frobenius covariants are


Sylvester's formula then states that


For instance, if f is defined by f(x) = x⁻¹, then Sylvester's formula computes the matrix inverse f(A) = A⁻¹ as