Sylvester's formula

Conditions

Sylvester's formula applies for any diagonalizable matrix A with k distinct eigenvalues, λ1, ..., λk, 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 mi 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 , then Sylvester's formula expresses the matrix inverse 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 Ai 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_{+}+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{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) , pp 101-103

References

1. 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.
2. Buchheim, Arthur. (1884). "On the Theory of Matrices". Proceedings of the London Mathematical Society.
3. Schwerdtfeger, Hans. (1938). "Les fonctions de matrices: Les fonctions univalentes. I, Volume 1". Hermann.