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 $A$. It states that
 * $$ f(A) = \sum_{i=1}^k f(\lambda_i) ~A_i ~,$$

where the $λ_{i}$ are the eigenvalues of $A$, and the matrices
 * $$ A_i \equiv \prod_{j=1 \atop j \ne i}^k \frac{1}{\lambda_i - \lambda_j} \left(A - \lambda_j I\right) $$

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 $λ_{i}$.

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

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

where $A$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 $$\plusmn 1$$, 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 $f$,
 * $$\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 $$ A =e^{i\frac{\pi}{2}(I-A)}=e^{-i\frac{\pi}{2}(I-A)}$$.