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 $A$ .^[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 $A$ , 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, $λ$ ₁, …, 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 2 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.

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

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