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]

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}\equiv \prod _{j=1 \atop j\neq i}^{k}{\frac {1}{\lambda _{i}-\lambda _{j}}}\left(A-\lambda _{j}I\right)

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, $λ$ 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}{\frac {1}{7}}&{\frac {1}{7}}\end{bmatrix}}={\begin{bmatrix}{\frac {3}{7}}&{\frac {3}{7}}\\{\frac {4}{7}}&{\frac {4}{7}}\end{bmatrix}}={\frac {A+2I}{5-(-2)}}\\A_{2}&=c_{2}r_{2}={\begin{bmatrix}{\frac {1}{7}}\\-{\frac {1}{7}}\end{bmatrix}}{\begin{bmatrix}4&-3\end{bmatrix}}={\begin{bmatrix}{\frac {4}{7}}&-{\frac {3}{7}}\\-{\frac {4}{7}}&{\frac {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}{\frac {3}{7}}&{\frac {3}{7}}\\{\frac {4}{7}}&{\frac {4}{7}}\end{bmatrix}}-{\frac {1}{2}}{\begin{bmatrix}{\frac {4}{7}}&-{\frac {3}{7}}\\-{\frac {4}{7}}&{\frac {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 Arthur Buchheim, based on Hermite interpolating polynomials, covers the general case:^[4]

f(A)=\sum _{i=1}^{s}\left[\sum _{j=0}^{n_{i}-1}{\frac {1}{j!}}\phi _{i}^{(j)}(\lambda _{i})\left(A-\lambda _{i}I\right)^{j}\prod _{j=1,j\neq i}^{s}\left(A-\lambda _{j}I\right)^{n_{j}}\right]

,

where $\phi _{i}(t):=f(t)/\prod _{j\neq i}\left(t-\lambda _{j}\right)^{n_{j}}$ .

A concise form is further given by Hans Schwerdtfeger,^[5]

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 $A$ _i are the corresponding Frobenius covariants of $A$

Special case

If a matrix $A$ is both Hermitian and unitary, then it can only have eigenvalues of $\pm 1$ , 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 $f$ ,

{\begin{aligned}f(\theta A)&=f(\theta )A_{+1}+f(-\theta )A_{-1}\\&=f(\theta ){\frac {I+A}{2}}+f(-\theta ){\frac {I-A}{2}}\\&={\frac {f(\theta )+f(-\theta )}{2}}I+{\frac {f(\theta )-f(-\theta )}{2}}A\\\end{aligned}}.

In particular, $e^{i\theta A}=(\cos \theta )I+(i\sin \theta )A$ and $A=e^{i{\frac {\pi }{2}}(I-A)}=e^{-i{\frac {\pi }{2}}(I-A)}$ .

References

^ ^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.

[horn-1] / Roger A. Horn and Charles R. Johnson (1991), Topics in Matrix Analysis. Cambridge University Press, ISBN 978-0-521-46713-1

[claer-2] 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.

[3] 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.

[4] 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.

[5] Schwerdtfeger, Hans (1938). Les fonctions de matrices: Les fonctions univalentes. I, Volume 1. Paris, France: Hermann.

[1]

[2]

[3]

[4]

[5]