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 of a matrix as a polynomial in , in terms of the eigenvalues and eigenvectors of ./
Roger A. Horn and Charles R. Johnson (1991), ''Topics in Matrix Analysis''. Cambridge University Press,
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.
It states that
:$f(A) = \sum_^k f(\lambda_i) ~A_i ~,$

where the are the eigenvalues of , and the matrices
: $A_i \equiv \prod_^k \frac \left(A - \lambda_j I\right)$
are the corresponding Frobenius covariants of , which are (projection) matrix Lagrange polynomials of .

Conditions

Sylvester's formula applies for any diagonalizable matrix with distinct eigenvalues, ₁, ..., _''k'', and any function defined on some subset of the complex numbers such that is well defined. The last condition means that every eigenvalue is in the domain of , and that every eigenvalue with multiplicity _''i'' > 1 is in the interior of the domain, with being () times differentiable at .

Example

Consider the two-by-two matrix:
:$A = \begin 1 & 3 \\ 4 & 2 \end.$

This matrix has two eigenvalues, 5 and −2. Its Frobenius covariants are
:$\begin A_1 &= c_1 r_1 = \begin 3 \\ 4 \end \begin \frac & \frac \end = \begin \frac & \frac \\ \frac & \frac \end = \frac\\ A_2 &= c_2 r_2 = \begin \frac \\ -\frac \end \begin 4 & -3 \end = \begin \frac & -\frac \\ -\frac & \frac \end = \frac. \end$

Sylvester's formula then amounts to
:$f(A) = f(5) A_1 + f(-2) A_2. \,$

For instance, if is defined by , then Sylvester's formula expresses the matrix inverse as
:$\frac \begin \frac & \frac \\ \frac & \frac \end - \frac \begin \frac & -\frac \\ -\frac & \frac \end = \begin -0.2 & 0.3 \\ 0.4 & -0.1 \end.$

Generalization

Sylvester's formula is only valid for diagonalizable matrices; an extension due to Arthur Buchheim, based on Hermite interpolating polynomials, covers the general case:
:f(A) = \sum_^ \left \sum_^ \frac \phi_i^(\lambda_i)\left(A - \lambda_i I\right)^j \prod_^\left(A - \lambda_j I\right)^ \right/math>,
where $\phi_i(t) := f(t)/\prod_\left(t - \lambda_j\right)^$.

A concise form is further given by Hans Schwerdtfeger,
:$f(A)=\sum_^ A_ \sum_^ \frac(A-\lambda_iI)^$,
where _''i'' are the corresponding Frobenius covariants of

Special case

If a matrix is both Hermitian and unitary, then it can only have eigenvalues of $\plusmn 1$, and therefore $A=A_+-A_-$, where $A_+$ is the projector onto the subspace with eigenvalue +1, and $A_-$ is the projector onto the subspace with eigenvalue $- 1$; By the completeness of the eigenbasis, $A_++A_-=I$. Therefore, for any analytic function ,
:$\begin f(\theta A)&=f(\theta)A_+f(-\theta)A_ \\ &=f(\theta)\frac+f(-\theta)\frac\\ &=\fracI+\fracA\\ \end .$

In particular, $e^=(\cos \theta)I+(i\sin \theta) A$ and $A =e^=e^$.

See also

* Adjugate matrix
* Holomorphic functional calculus
* Resolvent formalism

References

* F.R. Gantmacher, ''The Theory of Matrices'' v I (Chelsea Publishing, NY, 1960) , pp 101-103
*
*{{cite journal , last= Merzbacher , first= E , title = Matrix methods in quantum mechanics, journal= Am. J. Phys., volume= 36 , issue= 9 , pages= 814–821, year =1968, doi= 10.1119/1.1975154, bibcode= 1968AmJPh..36..814M
Matrix theory