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 λ_i are the eigenvalues of A, and the matrices A_i 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

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.^[1]^:Def.6.4.

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

${\begin{aligned}A_{1}&=c_{1}r_{1}={\begin{bmatrix}3\\4\end{bmatrix}}{\begin{bmatrix}1/7&1/7\end{bmatrix}}={\begin{bmatrix}3/7&3/7\\4/7&4/7\end{bmatrix}}={\frac {A+2I}{5-(-2)}}\\A_{2}&=c_{2}r_{2}={\begin{bmatrix}1/7\\-1/7\end{bmatrix}}{\begin{bmatrix}4&-3\end{bmatrix}}={\begin{bmatrix}4/7&-3/7\\-4/7&3/7\end{bmatrix}}={\frac {A-5I}{-2-5}}.\end{aligned}}$

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

↑ 1.0 1.1 Roger A. Horn and Charles R. Johnson (1991), Topics in Matrix Analysis. Cambridge University Press, ISBN 978-0-521-46713-1
↑ 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.

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