Sylvester's formula

Template:Short description In matrix theory, Sylvester's formula or Sylvester's matrix theorem (named after J. J. Sylvester) or Lagrange−Sylvester interpolation expresses an analytic function Template:Math of a matrix Template:Mvar as a polynomial in Template:Mvar, in terms of the eigenvalues and eigenvectors of Template:Mvar. It states that

$f(A)=\sum _{i=1}^{k}f({\lambda }_i)A_i,$

where the Template:Math are the eigenvalues of Template:Mvar, and the matrices

$A_i\equiv \prod _{\genfrac{}{}{0ex}{}{j=1}{j\ne i}}^k\frac{1}{{\lambda }_i-{\lambda }_j}(A-{\lambda }_{j}I)$

are the corresponding Frobenius covariants of Template:Mvar, which are (projection) matrix Lagrange polynomials of Template:Mvar.

Conditions

Sylvester's formula applies for any diagonalizable matrix Template:Mvar with Template:Mvar distinct eigenvalues, Template:Mvar₁, ..., Template:Mvar_k, and any function Template:Mvar defined on some subset of the complex numbers such that Template:Math is well defined. The last condition means that every eigenvalue Template:Math is in the domain of Template:Mvar, and that every eigenvalue Template:Math with multiplicity Template:Mvar_i > 1 is in the interior of the domain, with Template:Mvar being (Template:Math) times differentiable at Template:Math.Template:Rp

Example

Consider the two-by-two matrix:

$A=\left[\begin{array}{cc}1& 3\\ 4& 2\end{array}\right].$

This matrix has two eigenvalues, 5 and −2. Its Frobenius covariants are

$\begin{array}{cc}{A}_1& =c_1{r}_1=\left[\begin{array}{c}3\\ 4\end{array}\right]\left[\begin{array}{cc}\frac{1}{7}& \frac{1}{7}\end{array}\right]=\left[\begin{array}{cc}\frac{3}{7}& \frac{3}{7}\\ \frac{4}{7}& \frac{4}{7}\end{array}\right]=\frac{A+2I}{5-(-2)}\\ A_2& =c_2{r}_2=\left[\begin{array}{c}\frac{1}{7}\\ -\frac{1}{7}\end{array}\right]\left[\begin{array}{cc}4& -3\end{array}\right]=\left[\begin{array}{cc}\frac{4}{7}& -\frac{3}{7}\\ -\frac{4}{7}& \frac{3}{7}\end{array}\right]=\frac{A-5I}{-2-5}.\end{array}$

Sylvester's formula then amounts to

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

For instance, if Template:Mvar is defined by Template:Math, then Sylvester's formula expresses the matrix inverse Template:Math as

$\frac{1}{5}\left[\begin{array}{cc}\frac{3}{7}& \frac{3}{7}\\ \frac{4}{7}& \frac{4}{7}\end{array}\right]-\frac{1}{2}\left[\begin{array}{cc}\frac{4}{7}& -\frac{3}{7}\\ -\frac{4}{7}& \frac{3}{7}\end{array}\right]=\left[\begin{array}{cc}-0.2& 0.3\\ 0.4& -0.1\end{array}\right].$

Generalization

Sylvester's formula is only valid for matrices with no repeated eigenvalues; an extension due to Arthur Buchheim, based on Hermite interpolating polynomials, covers the general case:

$f(A)=\sum _{i=1}^s[\sum _{j=0}^{n_i-1}\frac{1}{j!}{\varphi }_i^{(j)}({\lambda }_i){(A-{\lambda }_{i}I)}^j\prod _{j=1,j\ne i}^s{(A-{\lambda }_{j}I)}^{n_j}]$,

where ${\varphi }_i(t):=f(t)/\prod _{j\ne i}{(t-{\lambda }_j)}^{n_j}$.

A concise form is further given by Hans Schwerdtfeger,

$f(A)=\sum _{i=1}^s{A}_i\sum _{j=0}^{n_i-1}\frac{f^{(j)}({\lambda }_i)}{j!}(A-{\lambda }_{i}I{)}^j$,

where Template:Mvar_i are the corresponding generalized Frobenius covariants of Template:Mvar

Special case

Template:See also If a matrix Template:Mvar 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 Template:Mvar,

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

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

See also

• Adjugate matrix
• Holomorphic functional calculus
• Resolvent formalism

References

• F.R. Gantmacher, The Theory of Matrices v I (Chelsea Publishing, NY, 1960) Template:ISBN , pp 101-103
• Template:Cite book
• Template:Cite journal