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]

${\displaystyle f(A)=\sum _{i=1}^{k}f(\lambda _{i})~A_{i}~,}$

where the λ_i are the eigenvalues of A, and the matrices

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

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, λ₁, ..., 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:

${\displaystyle A={\begin{bmatrix}1&3\\4&2\end{bmatrix}}.}$

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

${\displaystyle {\begin{aligned}A_{1}&=c_{1}r_{1}={\begin{bmatrix}3\\4\end{bmatrix}}{\begin{bmatrix}{\frac {1}{7}}&{\frac {1}{7}}\end{bmatrix}}={\begin{bmatrix}{\frac {3}{7}}&{\frac {3}{7}}\\{\frac {4}{7}}&{\frac {4}{7}}\end{bmatrix}}={\frac {A+2I}{5-(-2)}}\\A_{2}&=c_{2}r_{2}={\begin{bmatrix}{\frac {1}{7}}\\-{\frac {1}{7}}\end{bmatrix}}{\begin{bmatrix}4&-3\end{bmatrix}}={\begin{bmatrix}{\frac {4}{7}}&-{\frac {3}{7}}\\-{\frac {4}{7}}&{\frac {3}{7}}\end{bmatrix}}={\frac {A-5I}{-2-5}}.\end{aligned}}}$

Sylvester's formula then amounts to

${\displaystyle f(A)=f(5)A_{1}+f(-2)A_{2}.\,}$

For instance, if f is defined by f(x) = x⁻¹, then Sylvester's formula expresses the matrix inverse f(A) = A⁻¹ as

${\displaystyle {\frac {1}{5}}{\begin{bmatrix}{\frac {3}{7}}&{\frac {3}{7}}\\{\frac {4}{7}}&{\frac {4}{7}}\end{bmatrix}}-{\frac {1}{2}}{\begin{bmatrix}{\frac {4}{7}}&-{\frac {3}{7}}\\-{\frac {4}{7}}&{\frac {3}{7}}\end{bmatrix}}={\begin{bmatrix}-0.2&0.3\\0.4&-0.1\end{bmatrix}}.}$

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]

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

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

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

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

where A_i are the corresponding Frobenius covariants of A

Special case

See also: Euler's formula

If a matrix A is both Hermitian and unitary, then it can only have eigenvalues of ${\displaystyle \pm 1}$, and therefore ${\displaystyle A=A_{+}-A_{-}}$, where ${\displaystyle A_{+}}$ is the projector onto the subspace with eigenvalue +1, and ${\displaystyle A_{-}}$ is the projector onto the subspace with eigenvalue ${\displaystyle -1}$; By the completeness of the eigenbasis, ${\displaystyle A_{+}+A_{-}=I}$. Therefore, for any analytic function f,

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

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

See also

• Adjugate matrix
• Holomorphic functional calculus
• Resolvent formalism