Sylvester's formula

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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.

Contents 1 Statement 2 Example 3 Notes 4 References 5 External links

[edit] Statement

Let A be a diagonalizable matrix with k distinct eigenvalues, λ₁, …, λ_k. The Frobenius covariants A_i are defined by

^[1]

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 ith column of S by c_i and the ith row of S⁻¹ by r_i; c_i and r_i are left and right eigenvectors of A). Then A_i = c_ir_i. If A has multiple eigenvalues then A_i = Σ_j c_jr_j, where the sum is over all rows and columns associated with the eigenvalue λ_i.^[2]

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.^[3] Then, Sylvester's formula states that

^[4]

[edit] 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

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

[edit] Notes

1. ^ Horn & Johnson, equation (6.1.36)
2. ^ Horn & Johnson, page 521
3. ^ Horn & Johnson, Definition 6.4
4. ^ Horn & Johnson, equation (6.2.39)

[edit] References

• Horn, Roger A. & Johnson, Charles R. (1991), Topics in Matrix Analysis, Cambridge University Press, ISBN 9780521467131 .

[edit] External links

• Jon F. Claerbout, Sylvester's matrix theorem, a section of Fundamentals of Geophysical Data Processing, published in 1976.