Sylvester's formula

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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 in terms of the eigenvalues and eigenvectors of A.[1][2] It states that

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

where the λi are the eigenvalues of A, and the matrices Ai are the corresponding Frobenius covariants of A, 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[edit]

Sylvester's formula applies for any diagonalizable matrix A with k distinct eigenvalues, λ1, …, λk, 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 λi is in the domain of f, and that every eigenvalue λi with multiplicity mi > 1 is in the interior of the domain, with f being (mi − 1) times differentiable at λi.[1]:Def.6.4.

Example[edit]

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[edit]

  1. ^ a b Roger A. Horn and Charles R. Johnson (1991), Topics in Matrix Analysis. Cambridge University Press, ISBN 978-0-521-46713-1
  2. ^ Jon F. Claerbout (1976), Sylvester's matrix theorem, a section of Fundamentals of Geophysical Data Processing. Online version at sepwww.stanford.edu, accessed on 2010-03-14.