Misplaced Pages

Sylvester's formula

Article snapshot taken from Wikipedia with creative commons attribution-sharealike license. Give it a read and then ask your questions in the chat. We can research this topic together.
Formula in matrix theory

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

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

A i ≡ ∏ j = 1 j ≠ i k 1 λ i − λ j ( A − λ j I ) {\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

This article needs attention from an expert in mathematics. The specific problem is: The discussion of eigenvalues with multiplicities greater than one seems to be unnecessary, as the matrix is assumed to have distinct eigenvalues. WikiProject Mathematics may be able to help recruit an expert. (June 2023)

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 λi is in the domain of f, and that every eigenvalue λi with multiplicity mi > 1 is in the interior of the domain, with f being (mi - 1) times differentiable at λi.

Example

Consider the two-by-two matrix:

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

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

A 1 = c 1 r 1 = [ 3 4 ] [ 1 7 1 7 ] = [ 3 7 3 7 4 7 4 7 ] = A + 2 I 5 − ( − 2 ) A 2 = c 2 r 2 = [ 1 7 − 1 7 ] [ 4 − 3 ] = [ 4 7 − 3 7 − 4 7 3 7 ] = A − 5 I − 2 − 5 . {\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

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

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

1 5 [ 3 7 3 7 4 7 4 7 ] − 1 2 [ 4 7 − 3 7 − 4 7 3 7 ] = [ − 0.2 0.3 0.4 − 0.1 ] . {\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:

f ( A ) = ∑ i = 1 s [ ∑ j = 0 n i − 1 1 j ! ϕ i ( j ) ( λ i ) ( A − λ i I ) j ∏ j = 1 , j ≠ i s ( A − λ j I ) n j ] {\displaystyle f(A)=\sum _{i=1}^{s}\left} ,

where ϕ i ( t ) := f ( t ) / ∏ j ≠ i ( t − λ j ) n j {\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,

f ( A ) = ∑ i = 1 s A i ∑ j = 0 n i − 1 f ( j ) ( λ i ) j ! ( A − λ i I ) j {\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 Ai 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 ± 1 {\displaystyle \pm 1} , and therefore A = A + − A − {\displaystyle A=A_{+}-A_{-}} , where A + {\displaystyle A_{+}} is the projector onto the subspace with eigenvalue +1, and A − {\displaystyle A_{-}} is the projector onto the subspace with eigenvalue − 1 {\displaystyle -1} ; By the completeness of the eigenbasis, A + + A − = I {\displaystyle A_{+}+A_{-}=I} . Therefore, for any analytic function f,

f ( θ A ) = f ( θ ) A + 1 + f ( − θ ) A − 1 = f ( θ ) I + A 2 + f ( − θ ) I − A 2 = f ( θ ) + f ( − θ ) 2 I + f ( θ ) − f ( − θ ) 2 A . {\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, e i θ A = ( cos ⁡ θ ) I + ( i sin ⁡ θ ) A {\displaystyle e^{i\theta A}=(\cos \theta )I+(i\sin \theta )A} and A = e i π 2 ( I − A ) = e − i π 2 ( I − A ) {\displaystyle A=e^{i{\frac {\pi }{2}}(I-A)}=e^{-i{\frac {\pi }{2}}(I-A)}} .

See also

References

  1. ^ / Roger A. Horn and Charles R. Johnson (1991), Topics in Matrix Analysis. Cambridge University Press, ISBN 978-0-521-46713-1
  2. Jon F. Claerbout (1976), Sylvester's matrix theorem, a section of Fundamentals of Geophysical Data Processing. Online version at sepwww.stanford.edu, accessed on 2010-03-14.
  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. 16 (100): 267–269. doi:10.1080/14786448308627430. ISSN 1941-5982.
  4. Buchheim, Arthur (1884). "On the Theory of Matrices". Proceedings of the London Mathematical Society. s1-16 (1): 63–82. doi:10.1112/plms/s1-16.1.63. ISSN 0024-6115.
  5. Schwerdtfeger, Hans (1938). Les fonctions de matrices: Les fonctions univalentes. I, Volume 1. Paris, France: Hermann.
Category: