Sylvester's formula

In matrix theory, Sylvester's formula or Sylvester's matrix theorem (named after J. J. Sylvester) or Lagrange−Sylvester interpolation expresses an analytic function f(A) of a matrix A as a polynomial in A, in terms of the eigenvalues and eigenvectors of A. It states that

$f(A) = \sum_{i=1}^k f(\lambda_i) ~ A_i ~,$

where the λ_i are the eigenvalues of A, and the matrices A_i are the corresponding Frobenius covariants of A, which are (projection) matrix Lagrange polynomials of A.

Sylvester's formula (1883) is only valid for diagonalizable matrices; an extension due to A. Buchheim (1886) covers the general case.

Conditions

Sylvester's formula applies for any diagonalizable 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:

$A = \begin{bmatrix} 1 & 3 \\ 4 & 2 \end{bmatrix}.$

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

$\begin{align} A_1 &= c_1 r_1 = \begin{bmatrix} 3 \\ 4 \end{bmatrix} \begin{bmatrix} 1/7 & 1/7 \end{bmatrix} = \begin{bmatrix} 3/7 & 3/7 \\ 4/7 & 4/7 \end{bmatrix} = \frac{A+2I}{5-(-2)}\\ A_2 &= c_2 r_2 = \begin{bmatrix} 1/7 \\ -1/7 \end{bmatrix} \begin{bmatrix} 4 & -3 \end{bmatrix} = \begin{bmatrix} 4/7 & -3/7 \\ -4/7 & 3/7 \end{bmatrix}=\frac{A-5I}{-2-5}. \end{align}$

Sylvester's formula then amounts to

$f(A) = f(5) A_1 + f(-2) A_2. \,$

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

$\frac{1}{5} \begin{bmatrix} 3/7 & 3/7 \\ 4/7 & 4/7 \end{bmatrix} - \frac{1}{2} \begin{bmatrix} 4/7 & -3/7 \\ -4/7 & 3/7 \end{bmatrix} = \begin{bmatrix} -0.2 & 0.3 \\ 0.4 & -0.1 \end{bmatrix}.$

References

• F.R. Gantmacher, The Theory of Matrices v I (Chelsea Publishing, NY, 1960) ISBN 0-8218-1376-5 , pp 101-103