“Frobenius covariant”的意思、由来-开放百科全书

词条

Frobenius covariant

释义

Formal definition
Computing the covariants
Example
References

In matrix theory, the Frobenius covariants of a square matrix {{mvar|A}} are special polynomials of it, namely projection matrices A_i associated with the eigenvalues and eigenvectors of {{mvar|A}}.^[1]{{rp|pp.403,437–8}} They are named after the mathematician Ferdinand Frobenius.

Each covariant is a projection on the eigenspace associated with the eigenvalue {{math|λ_i}}.

Frobenius covariants are the coefficients of Sylvester's formula, which expresses a function of a matrix {{math|f(A)}} as a matrix polynomial, namely a linear combination

of that function's values on the eigenvalues of {{mvar|A}}.

Formal definition

Let {{mvar|A}} be a diagonalizable matrix with eigenvalues λ₁, …, λ_k.

The Frobenius covariant {{math|A_i}}, for i = 1,…, k, is the matrix

It is essentially the Lagrange polynomial with matrix argument. If the eigenvalue λ_i is simple, then as an idempotent projection matrix to a one-dimensional subspace, {{math|A_i}} has a unit trace.

Computing the covariants

The Frobenius covariants of a matrix {{mvar|A}} can be obtained from any eigendecomposition {{math|A {{=}} SDS⁻¹}}, where {{mvar|S}} is non-singular and {{mvar|D}} is diagonal with {{math|D_i,i {{=}} λ_i}}.

If {{mvar|A}} has no multiple eigenvalues, then let c_i be the {{mvar|i}}th right eigenvector of {{mvar|A}}, that is, the {{mvar|i}}th column of {{mvar|S}}; and let r_i be the {{mvar|i}}th left eigenvector of {{mvar|A}}, namely the {{mvar|i}}th row of {{mvar|S}}⁻¹. Then {{math|A_i {{=}} c_i r_i}}.

If {{mvar|A}} has an eigenvalue λ_i appear multiple times, then {{math|A_i {{=}} Σ_j c_j r_j}}, where the sum is over all rows and columns associated with the eigenvalue λ_i.^[1]{{rp|p.521}}

Example

Consider the two-by-two matrix:

This matrix has two eigenvalues, 5 and −2; hence {{math| (A−5)(A+2)}}=0.

The corresponding eigen decomposition is

Hence the Frobenius covariants, manifestly projections, are

with

Note {{math|trA₁{{=}}trA₂{{=}}1}}, as required.

References

1. ^¹Roger A. Horn and Charles R. Johnson (1991), Topics in Matrix Analysis. Cambridge University Press, {{ISBN|978-0-521-46713-1}}

1 : Matrix theory

随便看

开放百科全书收录14589846条英语、德语、日语等多语种百科知识，基本涵盖了大多数领域的百科知识，是一部内容自由、开放的电子版国际百科全书。