Frobenius covariant: Difference between revisions

Content deleted Content added

Inline

Latest revision as of 07:48, 25 January 2024

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]^{: pp.403, 437–8} 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 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)~.

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, $A i$ has a unit trace.

Computing the covariants

Ferdinand Georg Frobenius (1849–1917), German mathematician. His main interests were elliptic functions differential equations, and later group theory.

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 $i$ th right eigenvector of $A$ , that is, the $i$ th column of $S$ ; and let r_i be the $i$ th left eigenvector of $A$ , namely the $i$ th row of $S$ ⁻¹. Then $A i = c i r i$ .

If $A$ has an eigenvalue λ_i appearing multiple times, 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; hence $(A - 5)(A + 2) = 0$ .

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{array}{rl}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{array}}

with

A_{1}A_{2}=0,\qquad A_{1}+A_{2}=I~.

Note $tr A 1 = tr A 2 = 1$ , as required.

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]

@@ Line 1: / Line 1: @@
-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>{{rp|pp.403,437-8}} They are named after the mathematician [[Ferdinand Georg Frobenius|Ferdinand Frobenius]].
+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>{{rp|pp.403,437–8}} They are named after the mathematician [[Ferdinand Georg Frobenius|Ferdinand Frobenius]].
 Each covariant is a [[projection (linear algebra)|projection]]  on the [[eigenvalue, eigenvector and eigenspace|eigenspace]] associated with the eigenvalue {{math|''λ''<sub>''i''</sub>}}.
@@ Line 6: / Line 6: @@
 ==Formal definition==
-Let {{mvar|A}} be a [[diagonalizable matrix]] with {{mvar|k}} distinct eigenvalues, ''λ''<sub>1</sub>, &hellip;, ''λ''<sub>''k''</sub>.
+Let {{mvar|A}} be a [[diagonalizable matrix]] with eigenvalues ''λ''<sub>1</sub>, ..., ''λ''<sub>''k''</sub>.
-The Frobenius covariant {{math|''A''<sub>''i''</sub>}}, for ''i'' = 1,&hellip;, ''k'', is the matrix
+The Frobenius covariant {{math|''A''<sub>''i''</sub>}}, for ''i'' = 1,..., ''k'', is the matrix
 :<math> A_i \equiv  \prod_{j=1 \atop j \ne i}^k \frac{1}{\lambda_i-\lambda_j} (A - \lambda_j I)~. </math>
 It is essentially the [[Lagrange polynomial]] with matrix argument. If the eigenvalue  ''λ''<sub>''i''</sub> is simple, then as an idempotent projection matrix to a one-dimensional subspace,  {{math|''A''<sub>''i''</sub>}} has a unit [[Trace (linear algebra)|trace]].
@@ Line 18: / Line 18: @@
 The Frobenius covariants of a matrix {{mvar|A}} can be obtained from any [[eigendecomposition]] {{math|''A'' {{=}} ''SDS''<sup>−1</sup>}}, where {{mvar|S}} is non-singular and {{mvar|D}} is diagonal with {{math|''D''<sub>''i'',''i''</sub> {{=}} ''λ''<sub>''i''</sub>}}.
-If {{mvar|A}} has no multiple eigenvalues, then let ''c''<sub>''i''</sub> be the {{mvar|i}}th left eigenvector of {{mvar|A}}, that is, the {{mvar|i}}th column of {{mvar|S}}; and let ''r''<sub>''i''</sub> be the {{mvar|i}}th right eigenvector of {{mvar|A}}, namely the {{mvar|i}}th row of {{mvar|S}}<sup>−1</sup>.  Then {{math|''A''<sub>''i''</sub> {{=}} ''c''<sub>''i''</sub> ''r''<sub>''i''</sub>}}.
+If {{mvar|A}} has no multiple eigenvalues, then let ''c''<sub>''i''</sub> be the {{mvar|i}}th right eigenvector of {{mvar|A}}, that is, the {{mvar|i}}th column of {{mvar|S}}; and let ''r''<sub>''i''</sub> be the {{mvar|i}}th left eigenvector of {{mvar|A}}, namely the {{mvar|i}}th row of {{mvar|S}}<sup>−1</sup>.  Then {{math|''A''<sub>''i''</sub> {{=}} ''c''<sub>''i''</sub> ''r''<sub>''i''</sub>}}.
-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}}
+If {{mvar|A}} has an eigenvalue ''λ''<sub>''i''</sub> appearing multiple times, 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==
@@ Line 26: / Line 26: @@
 Consider the two-by-two matrix:
 :<math> A = \begin{bmatrix} 1 & 3 \\ 4 & 2 \end{bmatrix}.</math>
-This matrix has two eigenvalues, 5 and −2; hence {{math| (''A''−5)(''A''+2)}}=0.
+This matrix has two eigenvalues, 5 and −2; hence {{math| (''A'' − 5)(''A'' + 2) {{=}} 0}}.
 The corresponding eigen decomposition is
@@ Line 36: / Line 36: @@
 \end{array} </math>
 with
-:<math>A_1 A_2 =0 , \qquad A_1+A_2= I ~.  </math>
+:<math>A_1 A_2 = 0 , \qquad A_1 + A_2 = I ~.</math>
-Note {{math|tr''A''<sub>1</sub>{{=}}tr''A''<sub>2</sub>{{=}}1}}, as required.
+Note {{math|tr{{nnbsp}}''A''<sub>1</sub> {{=}} tr{{nnbsp}}''A''<sub>2</sub> {{=}} 1}}, as required.
 ==References==