Sylvester's formula#Special case

In matrix theory, Sylvester's formula or Sylvester's matrix theorem (named after J. J. Sylvester) or Lagrange−Sylvester interpolation expresses an analytic function {{math|f(A)}} of a matrix {{mvar|A}} as a polynomial in {{mvar|A}}, in terms of the eigenvalues and eigenvectors of {{mvar|A}}./

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. [http://sepwww.stanford.edu/sep/prof/fgdp/c5/paper_html/node3.html Online version] at sepwww.stanford.edu, accessed on 2010-03-14.

It states that{{Cite journal|last=Sylvester|first=J.J.|date=1883|title=XXXIX. On the equation to the secular inequalities in the planetary theory|journal=The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science|language=en|volume=16|issue=100|pages=267–269|doi=10.1080/14786448308627430|issn=1941-5982|url=https://zenodo.org/record/2462638}}

: $f(A) = \sum_{i=1}^k f(\lambda_i) ~A_i ~,$

where the {{math|λ_i}} are the eigenvalues of {{mvar|A}}, and the matrices

: $A_i \equiv \prod_{j=1 \atop j \ne i}^k \frac{1}{\lambda_i - \lambda_j} \left(A - \lambda_j I\right)$

are the corresponding Frobenius covariants of {{mvar|A}}, which are (projection) matrix Lagrange polynomials of {{mvar|A}}.

Conditions

{{expert needed|mathematics|reason=The discussion of eigenvalues with multiplicities greater than one seems to be unnecessary, as the matrix is assumed to have distinct eigenvalues|date=June 2023}}

Sylvester's formula applies for any diagonalizable matrix {{mvar|A}} with {{mvar|k}} distinct eigenvalues, {{mvar|λ}}₁, ..., {{mvar|λ}}_k, and any function {{mvar|f}} defined on some subset of the complex numbers such that {{math|f(A)}} is well defined. The last condition means that every eigenvalue {{math|λ_i}} is in the domain of {{mvar|f}}, and that every eigenvalue {{math|λ_i}} with multiplicity {{mvar|m}}_i > 1 is in the interior of the domain, with {{mvar|f}} being ({{math|m_i - 1}}) times differentiable at {{math|λ_i}}.{{rp|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

: $\begin{align}
A_1 &= c_1 r_1 = \begin{bmatrix} 3 \\ 4 \end{bmatrix} \begin{bmatrix} \frac{1}{7} & \frac{1}{7} \end{bmatrix} = \begin{bmatrix} \frac{3}{7} & \frac{3}{7} \\ \frac{4}{7} & \frac{4}{7} \end{bmatrix} = \frac{A + 2I}{5 - (-2)}\\
A_2 &= c_2 r_2 = \begin{bmatrix} \frac{1}{7} \\ -\frac{1}{7} \end{bmatrix} \begin{bmatrix} 4 & -3 \end{bmatrix} = \begin{bmatrix} \frac{4}{7} & -\frac{3}{7} \\ -\frac{4}{7} & \frac{3}{7} \end{bmatrix} = \frac{A - 5I}{-2 - 5}.
\end{align}$

Sylvester's formula then amounts to

: $f(A) = f(5) A_1 + f(-2) A_2. \,$

For instance, if {{mvar|f}} is defined by {{math|f(x) {{=}} x⁻¹}}, then Sylvester's formula expresses the matrix inverse {{math|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:{{Cite journal|last=Buchheim|first=Arthur|date=1884|title=On the Theory of Matrices|journal=Proceedings of the London Mathematical Society|language=en|volume=s1-16|issue=1|pages=63–82|doi=10.1112/plms/s1-16.1.63|issn=0024-6115|url=https://zenodo.org/record/2131774}}

: $f(A) = \sum_{i=1}^{s} \left[ \sum_{j=0}^{n_{i}-1} \frac{1}{j!} \phi_i^{(j)}(\lambda_i)\left(A - \lambda_i I\right)^j \prod_{{j=1,j\ne i}}^{s}\left(A - \lambda_j I\right)^{n_j} \right]$ ,

where $\phi_i(t) := f(t)/\prod_{j\ne i}\left(t - \lambda_j\right)^{n_j}$ .

A concise form is further given by Hans Schwerdtfeger,{{Cite book|title=Les fonctions de matrices: Les fonctions univalentes. I, Volume 1|last=Schwerdtfeger|first=Hans|publisher=Hermann|year=1938|location=Paris, France}}

: $f(A)=\sum_{i=1}^{s} A_{i} \sum_{j=0}^{n_{i}-1} \frac{f^{(j)}(\lambda_i)}{j!}(A-\lambda_iI)^{j}$ ,

where {{mvar|A}}_i are the corresponding Frobenius covariants of {{mvar|A}}

Special case

If a matrix {{mvar|A}} 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 {{mvar|f}},

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

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

F.R. Gantmacher, The Theory of Matrices v I (Chelsea Publishing, NY, 1960) {{ISBN|0-8218-1376-5}} , pp 101-103
{{Cite book|title=Functions of matrices: theory and computation|last=Higham|first=Nicholas J.|date=2008|publisher=Society for Industrial and Applied Mathematics (SIAM)|isbn=9780898717778|location=Philadelphia|oclc=693957820}}
{{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 }}

Category:Matrix theory