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In matrix theory, the Frobenius covariants of a square matrix $A$ are special polynomials of it, namely projection matrices A_i associated with the eigenvalues and eigenvectors of $A$ .^[1] They are named after the mathematician Ferdinand Frobenius.

Each covariant is a projection on the eigenspace associated with the eigenvalue $λ i$ . Frobenius covariants are the coefficients of Sylvester's formula, which expresses a function of a matrix $f (A)$ as a matrix polynomial, namely a linear combination of that function's values on the eigenvalues of $A$ .

Formal definition

Let $A$ be a diagonalizable matrix with $k$ distinct eigenvalues, λ₁, …, λ_k. The Frobenius covariant A_i, for i = 1,…, k, is the matrix

A_{i}\equiv \prod _{j=1 \atop j\neq i}^{k}{\frac {1}{\lambda _{i}-\lambda _{j}}}(A-\lambda _{j}I).

Computing the covariants

The Frobenius covariants of a matrix A can be obtained from any eigendecomposition A = SDS⁻¹, where S is non-singular and D is diagonal with $D i, i = λ i$ . If A has no multiple eigenvalues, then let c_i be the ith left eigenvector of A, that is, the i th column of S; and let r_i be the i th right eigenvector of $A$ , namely the i th row of S⁻¹. 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.^[1]^: p.521

Example

Consider the two-by-two matrix:

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

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

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

Hence the Frobenius covariants, manifestly projections, are

{\begin{aligned}A_{1}&=c_{1}r_{1}={\begin{bmatrix}3\\4\end{bmatrix}}{\begin{bmatrix}1/7&1/7\end{bmatrix}}={\begin{bmatrix}3/7&3/7\\4/7&4/7\end{bmatrix}}=A_{1}^{2}\\A_{2}&=c_{2}r_{2}={\begin{bmatrix}1/7\\-1/7\end{bmatrix}}{\begin{bmatrix}4&-3\end{bmatrix}}={\begin{bmatrix}4/7&-3/7\\-4/7&3/7\end{bmatrix}}=A_{2}^{2}~.\end{aligned}}

References

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

[horn-1] Roger A. Horn and Charles R. Johnson (1991), Topics in Matrix Analysis. Cambridge University Press, ISBN 978-0-521-46713-1

[1]

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-In [[Matrix (mathematics)|matrix theory]], the '''Frobenius covariants''' of a [[square matrix]] ''A'' are matrices ''A''<sub>''i''</sub> associated with the [[eigenvalue, eigenvector and eigenspace|eigenvalues and eigenvectors]] of ''A''.<ref name=horn>
+In [[Matrix (mathematics)|matrix theory]], the '''Frobenius covariants''' of a [[square matrix]]  {{mvar|A}} are special polynomials of it, namely  [[projection (linear algebra)|projection]]  matrices ''A''<sub>''i''</sub> associated with the [[eigenvalue, eigenvector and eigenspace|eigenvalues and eigenvectors]] of  {{mvar|A}}.<ref name=horn>Roger A. Horn and Charles R. Johnson (1991), ''Topics in Matrix Analysis''. Cambridge University Press, ISBN 978-0-521-46713-1</ref>  They are named after the mathematician [[Ferdinand Georg Frobenius|Ferdinand Frobenius]].
-  Roger A. Horn and Charles R. Johnson (1991), ''Topics in Matrix Analysis''. Cambridge University Press, ISBN 978-0-521-46713-1
-</ref>  Each covariant is a [[projection (linear algebra)|projection]]<!--OF WHAT?--> on the [[eigenvalue, eigenvector and eigenspace|eigenspace]] associated with ''λ''<sub>''i''</sub>.
+Each covariant is a [[projection (linear algebra)|projection]]  on the [[eigenvalue, eigenvector and eigenspace|eigenspace]] associated with the eigenvalue {{math|''λ''<sub>''i''</sub>}}.
-Frobenius covariants are the coefficients of [[Sylvester's formula]], that expresses a [[matrix function|function of a matrix]] ''f''(''A'') as a linear combination of its values on the eigenvalues of ''A''. They are named after the mathematician [[Ferdinand Georg Frobenius|Ferdinand Frobenius]].
+Frobenius covariants are the coefficients of [[Sylvester's formula]], which expresses a [[matrix function|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 ''A'' be a [[diagonalizable matrix]] with ''k'' distinct eigenvalues, ''λ''<sub>1</sub>, &hellip;, ''λ''<sub>''k''</sub>. The Frobenius covariant ''A''<sub>''i''</sub>, for ''i'' = 1,&hellip;, ''k'', is the matrix
+Let {{mvar|A}} be a [[diagonalizable matrix]] with {{mvar|k}} distinct eigenvalues, ''λ''<sub>1</sub>, &hellip;, ''λ''<sub>''k''</sub>. The Frobenius covariant ''A''<sub>''i''</sub>, for ''i'' = 1,&hellip;, ''k'', is the matrix
-:<math> A_i = \prod_{j=1 \atop j \ne i}^k \frac{1}{\lambda_i-\lambda_j} (A - \lambda_j I). </math>
+:<math> A_i \equiv  \prod_{j=1 \atop j \ne i}^k \frac{1}{\lambda_i-\lambda_j} (A - \lambda_j I). </math>
 ==Computing the covariants==
-The Frobenius covariants of a matrix ''A'' can be obtained from any [[eigendecomposition]] ''A'' = ''SDS''<sup>−1</sup>, where ''S'' is non-singular and ''D'' is diagonal with ''D''<sub>''i'',''i''</sub> = ''λ''<sub>''i''</sub>.
+The Frobenius covariants of a matrix ''A'' can be obtained from any [[eigendecomposition]] ''A'' = ''SDS''<sup>−1</sup>, where ''S'' is non-singular and ''D'' is diagonal with {{math|''D''<sub>''i'',''i''</sub> {{=}} ''λ''<sub>''i''</sub>}}.
-If ''A'' has no multiple eigenvalues, then let ''c''<sub>''i''</sub> be the ''i''th left eigenvector of ''A'', that is, the ''i''th column of ''S''; and let ''r''<sub>''i''</sub> be the ''i''th right eigenvector of ''A'', namely the ''i''th row of ''S''<sup>−1</sup>.  Then ''A''<sub>''i''</sub> = ''c''<sub>''i''</sub>''r''<sub>''i''</sub>.
+If ''A'' has no multiple eigenvalues, then let ''c''<sub>''i''</sub> be the ''i''th left eigenvector of ''A'', that is, the ''i'' th column of ''S''; and let ''r''<sub>''i''</sub> be the ''i'' th right eigenvector of {{mvar|A}}, namely the ''i'' th row of ''S''<sup>−1</sup>.  Then ''A''<sub>''i''</sub> = ''c''<sub>''i''</sub> ''r''<sub>''i''</sub>.
-If ''A'' has multiple eigenvalues then ''A''<sub>''i''</sub> = Σ<sub>''j''</sub> ''c''<sub>''j''</sub>''r''<sub>''j''</sub>, where the sum is over all rows and columns associated with the eigenvalue ''λ''<sub>''i''</sub>.<ref name=horn/>{{rp|p.521}}
+If {{mvar|A}} has multiple eigenvalues, then {{math|''A''<sub>''i''</sub> {{=}} Σ<sub>''j''</sub> ''c''<sub>''j''</sub> ''r''<sub>''j''</sub>}}, where the sum is over all rows and columns associated with the eigenvalue ''λ''<sub>''i''</sub>.<ref name=horn/>{{rp|p.521}}
 ==Example==