Sylvester's formula

In matrix theory, Sylvester's formula, named after James Joseph Sylvester, expresses matrix functions in terms of the eigenvalues and eigenvectors of a matrix. It is only valid for diagonalizable matrices; an extension due to Buchheim covers the general case.

Statement

Let "A" be a diagonalizable matrix with "k" distinct eigenvalues, λ₁, …, λ_"k". The Frobenius covariants "A"_"i" are defined by:$A\_i\; =\; prod\_\{j=1\; atop\; j\; e\; i\}^k\; frac\{1\}\{lambda\_i-lambda\_j\}\; (A\; -\; lambda\_j\; I).$ [Horn & Johnson, equation (6.1.36)] The Frobenius covariants "A"_"i" are projections on the eigenspace associated with λ_"i". They can be found by diagonalizing the matrix "A". using the eigendecomposition "A" = "SDS"⁻¹. First, suppose that "A" has no multiple eigenvalues. Let "A" = "SDS"⁻¹ be the eigendecomposition of "A" where the diagonal matrix "D" has the eigenvalues λ₁, …, λ_"k" (in that order) on the diagonal. Denote the "i"th column of "S" by "c"_"i" and the "i"th row of "S"⁻¹ by "r"_"i"; "c"_"i" and "r"_"i" are left and right eigenvectors of "A"). Then "A"_"i" = "c"_"i""r"_"i". If "A" has multiple eigenvalues then "A"_"i" = Σ_"j" "c"_"j""r"_"j", where the sum is over all rows and columns associated with the eigenvalue λ_"i". [Horn & Johnson, page 521]

Now, let "f": "D" → C with "D" ⊂ C be a function for which "f"("A") is well defined; this means that every eigenvalue λ_"i" is in the domain "D" and that every λ_"i" with multiplicity "m"_"i" is in the interior of the domain with "f" ("m"_"i" − 1) times differentiable at λ_"i". [Horn & Johnson, Definition 6.4] Then, Sylvester's formula states that:$f(A)\; =\; sum\_\{i=1\}^k\; f(lambda\_i)\; A\_i.$ [Horn & Johnson, equation (6.2.39)]

Example

Consider the two-by-two matrix::This matrix has two eigenvalues, 5 and −2. Its eigendecomposition is:Hence the Frobenius covariants are:Sylvester's formula states that:$f(A)\; =\; f(5)\; A\_1\; +\; f(-2)\; A\_2.\; ,$For instance, if "f" is defined by "f"("x") = "x"⁻¹, then Sylvester's formula computes the matrix inverse "f"("A") = "A"⁻¹ as:

Notes

References

* citation | first1=Roger A. | last1=Horn | first2=Charles R. | last2=Johnson | year=1991 | title=Topics in Matrix Analysis | publisher=Cambridge University Press | isbn=9780521467131 .

External links

* Jon F. Claerbout, [http://sepwww.stanford.edu/ftp/prof/fgdp/c5/paper_html/node3.html Sylvester's matrix theorem] , a section of "Fundamentals of Geophysical Data Processing", published in 1976.