matrix theory In mathematics, a matrix (: matrices) is a rectangular array or table of numbers, symbols, or expressions, with elements or entries arranged in rows and columns, which is used to represent a mathematical object or property of such an object. ...

, the Frobenius covariants of a

square matrix In mathematics, a square matrix is a Matrix (mathematics), matrix with the same number of rows and columns. An ''n''-by-''n'' matrix is known as a square matrix of order Any two square matrices of the same order can be added and multiplied. Squ ...

are special polynomials of it, namely

projection Projection or projections may refer to: Physics * Projection (physics), the action/process of light, heat, or sound reflecting from a surface to another in a different direction * The display of images by a projector Optics, graphics, and carto ...

matrices ''A''_''i'' associated with the

eigenvalues and eigenvectors In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by a ...

of .Roger A. Horn and Charles R. Johnson (1991), ''Topics in Matrix Analysis''. Cambridge University Press, They are named after the mathematician Ferdinand Frobenius. Each covariant is a

on the

eigenspace In linear algebra, an eigenvector ( ) or characteristic vector is a Vector (mathematics and physics), vector that has its direction (geometry), direction unchanged (or reversed) by a given linear map, linear transformation. More precisely, an e ...

associated with the eigenvalue . Frobenius covariants are the coefficients of Sylvester's formula, which expresses a function of a matrix as a matrix polynomial, namely a linear combination of that function's values on the eigenvalues of .

Formal definition

Let be a

diagonalizable matrix In linear algebra, a square matrix A is called diagonalizable or non-defective if it is matrix similarity, similar to a diagonal matrix. That is, if there exists an invertible matrix P and a diagonal matrix D such that . This is equivalent to ...

with eigenvalues ''λ''₁, ..., ''λ''_''k''. The Frobenius covariant , for ''i'' = 1,..., ''k'', is the matrix :

A_i \equiv  \prod_^k \frac (A - \lambda_j I)~.

It is essentially the

Lagrange polynomial In numerical analysis, the Lagrange interpolating polynomial is the unique polynomial of lowest degree that interpolates a given set of data. Given a data set of coordinate pairs (x_j, y_j) with 0 \leq j \leq k, the x_j are called ''nodes'' ...

with matrix argument. If the eigenvalue ''λ''_''i'' is simple, then as an idempotent projection matrix to a one-dimensional subspace, has a unit trace.

Computing the covariants

The Frobenius covariants of a matrix can be obtained from any eigendecomposition , where is non-singular and is diagonal with . If has no multiple eigenvalues, then let ''c''_''i'' be the th right eigenvector of , that is, the th column of ; and let ''r''_''i'' be the th left eigenvector of , namely the th row of ⁻¹. Then . If has an eigenvalue ''λ''_''i'' appearing multiple times, then , where the sum is over all rows and columns associated with the eigenvalue ''λ''_''i''.

Example

Consider the two-by-two matrix: :

A = \begin 1 & 3 \\ 4 & 2 \end.

This matrix has two eigenvalues, 5 and −2; hence . The corresponding eigen decomposition is :

A = \begin 3 & 1/7 \\ 4 & -1/7 \end \begin 5 & 0 \\ 0 & -2 \end \begin 3 & 1/7 \\ 4 & -1/7 \end^ = \begin 3 & 1/7 \\ 4 & -1/7 \end \begin 5 & 0 \\ 0 & -2 \end \begin 1/7 & 1/7 \\ 4 & -3 \end.

Hence the Frobenius covariants, manifestly projections, are :

\begin
A_1 &= c_1 r_1 = \begin 3 \\ 4 \end \begin 1/7 & 1/7 \end = \begin 3/7 & 3/7 \\ 4/7 & 4/7 \end = A_1^2\\
A_2 &= c_2 r_2 = \begin 1/7 \\ -1/7 \end \begin 4 & -3 \end = \begin 4/7 & -3/7 \\ -4/7 & 3/7 \end=A_2^2 ~,
\end

with :

A_1 A_2 = 0 , \qquad A_1 + A_2 = I ~.

Note , as required.

References

{{Reflist Matrix theory