Sylvester's formula explained

In matrix theory, Sylvester's formula or Sylvester's matrix theorem (named after J. J. Sylvester) or Lagrange−Sylvester interpolation expresses an analytic function of a matrix as a polynomial in, in terms of the eigenvalues and eigenvectors of .^{[1] [2]} It states that^[3]

f(A) = \sum ^{k} _{i =1} f( λ _{i)} ~A _{i} ~,

where the are the eigenvalues of, and the matrices

A _{i} \equiv \prod ^{k} _{j =1 \atop j \ne i} (1)/(λ _{i} -λ _{j}) \left(A - λ _{j} I\right) are the corresponding Frobenius covariants of, which are (projection) matrix Lagrange polynomials of .

Conditions

Sylvester's formula applies for any diagonalizable matrix with distinct eigenvalues, ₁, ..., _k, and any function defined on some subset of the complex numbers such that is well defined. The last condition means that every eigenvalue is in the domain of, and that every eigenvalue with multiplicity _i > 1 is in the interior of the domain, with being times differentiable at .^[1]

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} (1)/(7) & (1)/(7) \end{bmatrix} = \begin{bmatrix} (3)/(7) & (3)/(\ 7) (4)/(7) & (4)/(7) \end{bmatrix} = (A +2 I)/(5- ( -2 )) \\ A _{2} & = c _{2} r _{2} = \begin{bmatrix} (1)/(\ - 7) (1)/(7) \end{bmatrix} \begin{bmatrix} 4 & -3 \end{bmatrix} = \begin{bmatrix} (4)/(7) & - (3)/(\ - 7) (4)/(7) & (3)/(7) \end{bmatrix} = (A -5 I)/(-2-5) . \end{align}

Sylvester's formula then amounts to

f(A) = f( 5 ) A _{1} + f( -2 ) A _{2.}

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

(1)/(5) \begin{bmatrix} (3)/(7) & (3)/(\ 7) (4)/(7) & (4)/(7) \end{bmatrix} - (1)/(2) \begin{bmatrix} (4)/(7) & - (3)/(\ - 7) (4)/(7) & (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]

f(A) = \sum ^{s} _{i =1} \left[ \sum ^{n _{i} -1} _{j =0} (1)/(j!) \phi ^{(j)} _{i} ( λ _{i)\left(A} - λ _{i} I\right) ^{j} \prod _{{j=1,j\ne} }^\left(A - \lambda_j I\right)^ \right],where \phi _{i(t)} : = f(t)/\prod _{j\ne} \left(t - λ ^{n _{j}} _{j\right)} .

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

f(A) = \sum ^{s} _{i =1} A _{i} \sum ^{n _{i} -1} _{j =0} (f ^{(j)} ( λ _{i)})/(j!) (A -λ ^{j} _{iI)} ,where _i are the corresponding Frobenius covariants of

Special case

See also: Euler's formula. If a matrix 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,\begin{align} f(\theta A)& = f(\theta)A _{+1} + f( - \theta)A _{-1} \\ & = f(\theta) (I + A)/(+ f( - \theta) 2) (I - A)/(\\ & = 2) (f(\theta) + f( - \theta))/(I + 2) (f(\theta) - f( - \theta))/(2) A\\ \end{align} .

In particular,

e ^{i\theta} = (\cos \theta)I + (i\sin \theta) A and A = e ^{i (\pi)/((I - A) 2)} = e ^{- i (\pi)/((I - A) 2)} .

See also

• Adjugate matrix
• Holomorphic functional calculus
• Resolvent formalism

References

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

Notes and References

1. / Roger A. Horn and Charles R. Johnson (1991), Topics in Matrix Analysis. Cambridge University Press,
2. [Jon Claerbout|Jon F. Claerbout]
3. 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. en. 16. 100. 267–269. 10.1080/14786448308627430. 1941-5982.
4. Buchheim. Arthur. 1884. On the Theory of Matrices. Proceedings of the London Mathematical Society. en. s1-16. 1. 63–82. 10.1112/plms/s1-16.1.63. 0024-6115.
5. Book: Schwerdtfeger, Hans. Les fonctions de matrices: Les fonctions univalentes. I, Volume 1. Hermann. 1938. Paris, France.