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[3]
where the
are the corresponding Frobenius covariants of A, which are (projection) matrix Lagrange polynomials of A.
Conditions
[edit]This article needs attention from an expert in mathematics. The specific problem is: The discussion of eigenvalues with multiplicities greater than one seems to be unnecessary, as the matrix is assumed to have distinct eigenvalues.(June 2023) |
Sylvester's formula applies for any diagonalizable matrix A with k distinct eigenvalues,
Example
[edit]Consider the two-by-two matrix:
This matrix has two eigenvalues, 5 and −2. Its Frobenius covariants are
Sylvester's formula then amounts to
For instance, if f is defined by f(x) = x−1, then Sylvester's formula expresses the matrix inverse f(A) = A−1 as
Generalization
[edit]Sylvester's formula is only valid for diagonalizable matrices; an extension due to Arthur Buchheim, based on Hermite interpolating polynomials, covers the general case:[4]
- ,
where .
A concise form is further given by Hans Schwerdtfeger,[5]
- ,
where Ai are the corresponding Frobenius covariants of A
Special case
[edit]If a matrix A is both Hermitian and unitary, then it can only have eigenvalues of , and therefore , where is the projector onto the subspace with eigenvalue +1, and is the projector onto the subspace with eigenvalue ; By the completeness of the eigenbasis, . Therefore, for any analytic function f,
In particular, and .
See also
[edit]References
[edit]- ^ a b / 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, Arthur (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. Bibcode:1968AmJPh..36..814M. doi:10.1119/1.1975154.