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

[math]\displaystyle{ f(A) = \sum_{i=1}^k f(\lambda_i) ~A_i ~, }[/math]

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

[math]\displaystyle{ A_i \equiv \prod_{j=1 \atop j \ne i}^k \frac{1}{\lambda_i - \lambda_j} \left(A - \lambda_j I\right) }[/math]

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.

Example

Consider the two-by-two matrix:

[math]\displaystyle{ A = \begin{bmatrix} 1 & 3 \\ 4 & 2 \end{bmatrix}. }[/math]

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

[math]\displaystyle{ \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} }[/math]

Sylvester's formula then amounts to

[math]\displaystyle{ f(A) = f(5) A_1 + f(-2) A_2. \, }[/math]

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

[math]\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}. }[/math]

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:

[math]\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\ne i}}^{s}\left(A - \lambda_j I\right)^{n_j} \right] }[/math],

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

A concise form is further given by Hans Schwerdtfeger,

[math]\displaystyle{ f(A)=\sum_{i=1}^{s} A_{i} \sum_{j=0}^{n_{i}-1} \frac{f^{(j)}(\lambda_i)}{j!}(A-\lambda_iI)^{j} }[/math],

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 [math]\displaystyle{ \plusmn 1 }[/math], and therefore [math]\displaystyle{ A=A_+-A_- }[/math], where [math]\displaystyle{ A_+ }[/math] is the projector onto the subspace with eigenvalue +1, and [math]\displaystyle{ A_- }[/math] is the projector onto the subspace with eigenvalue [math]\displaystyle{ - 1 }[/math]; By the completeness of the eigenbasis, [math]\displaystyle{ A_++A_-=I }[/math]. Therefore, for any analytic function f,

[math]\displaystyle{ \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} . }[/math]

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

See also

• Adjugate matrix
• Holomorphic functional calculus
• Resolvent formalism

References

• F.R. Gantmacher, The Theory of Matrices v I (Chelsea Publishing, NY, 1960) ISBN:0-8218-1376-5 , pp 101-103
• Higham, Nicholas J. (2008). Functions of matrices: theory and computation. Philadelphia: Society for Industrial and Applied Mathematics (SIAM). ISBN 9780898717778. OCLC 693957820. 
• Merzbacher, E (1968). "Matrix methods in quantum mechanics". Am. J. Phys. 36 (9): 814–821. doi:10.1119/1.1975154. Bibcode: 1968AmJPh..36..814M.