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[3]
\( f(A)=\sum _{i=1}^{k}f(\lambda _{i})~A_{i}~, \)
where the λi are the eigenvalues of A, and the matrices
\( {\displaystyle A_{i}\equiv \prod _{j=1 \atop j\neq i}^{k}{\frac {1}{\lambda _{i}-\lambda _{j}}}(A-\lambda _{j}I)}\)
are the corresponding Frobenius covariants of A, which are (projection) matrix Lagrange polynomials of A.
Conditions
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
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
\( {\frac {1}{5}}{\begin{bmatrix}3/7&3/7\\4/7&4/7\end{bmatrix}}-{\frac {1}{2}}{\begin{bmatrix}4/7&-3/7\\-4/7&3/7\end{bmatrix}}={\begin{bmatrix}-0.2&0.3\\0.4&-0.1\end{bmatrix}}.\)
Generalization
Sylvester's formula is only valid for diagonalizable matrices; an extension due to A. Buchheim, based on Hermite interpolating polynomials, covers the general case:[4]
\( {\displaystyle f(A)=\sum _{i=1}^{s}\left[\sum _{j=0}^{n_{i}-1}{\frac {1}{j!}}\phi _{i}^{(j)}(\lambda _{i})(A-\lambda _{i}I)^{j}\prod _{j=1,j\neq i}^{s}(A-\lambda _{j}I)^{n_{j}}\right]},\)
where \( {\displaystyle \phi _{i}(t):=f(t)/\prod _{j\neq i}(t-\lambda _{j})^{n_{j}}}.\)
A concise form is further given by Schwerdtfeger,[5]
\( {\displaystyle f(A)=\sum _{i=1}^{s}A_{i}\sum _{j=0}^{n_{i}-1}{\frac {f^{(j)}(\lambda _{i})}{j!}}(A-\lambda _{i}I)^{j}},\)
where Ai are the corresponding Frobenius covariants of A
See also
Adjugate matrix
Holomorphic functional calculus
Resolvent formalism
References
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.
Sylvester, J.J. (1883). "XXXIX. On the equation to the secular inequalities in the planetary theory". The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science. 16 (100): 267–269. doi:10.1080/14786448308627430. ISSN 1941-5982.
Buchheim, A. (1884). "On the Theory of Matrices". Proceedings of the London Mathematical Society. s1-16 (1): 63–82. doi:10.1112/plms/s1-16.1.63. ISSN 0024-6115.
Schwerdtfeger, Hans (1938). Les fonctions de matrices: Les fonctions univalentes. I, Volume 1. Paris, France: Hermann.
F.R. Gantmacher, The Theory of Matrices v I (Chelsea Publishing, NY, 1960) ISBN 0-8218-1376-5 , pp 101-103
Higham, Nicholas J. (2008). Functions of matrices: theory and computation. Philadelphia: Society for Industrial and Applied Mathematics (SIAM). ISBN 9780898717778. OCLC 693957820.
Merzbacher, E (1968). "Matrix methods in quantum mechanics". Am. J. Phys. 36 (9): 814–821. doi:10.1119/1.1975154.
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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