Permutation polynomial: Difference between revisions

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{{distinguish|Polynomial matrix}}
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In mathematics, a '''matrix polynomial''' is a polynomial with [[matrix (mathematics)|matrices]] as variablesExamples include:


::<math>P(A) = \sum_{i=0}^n{ a_i A^i} =a_0 I + a_1 A + a_2 A^2 + \cdots + a_n A^n,</math>
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: where ''P'' is a polynomial,
:: <math>P(x) = \sum_{i=0}^n{ a_i x^i} =a_0  + a_1 x+ a_2 x^2 + \cdots + a_n x^n,</math>
: and ''I'' is the identity matrix.
:: <math>\left[A,B\right] = A B - B A,</math>
: the [[commutator#Ring theory|commutator]] of ''A'' and ''B''.
 
A '''matrix polynomial equation''' is an equality between two matrix polynomials, which holds for the specific matrices in question. If <math>P(A) = Q(A)</math>, (where ''A'' is a matrix over a field), then the [[eigenvalues]] of ''A'' satisfy the '''characteristic equation'''{{Disputed-inline|Characteristic equation|date=October 2010}} <math>P(\lambda) = Q(\lambda)</math>.<br/>
A '''matrix polynomial identity''' is a matrix polynomial equation which holds for all matricies ''A'' in a specified [[matrix ring]] ''M<sub>n</sub>''(''R'').
 
==Matrix geometrical series==
 
Matrix polynomials can be used to sum a matrix geometrical series as one would an ordinary [[geometric series]],
 
:<math>S=I+A+A^2+\cdots +A^n</math>
:<math>AS=A+A^2+A^3+\cdots +A^{n+1}</math>
:<math>(I-A)S=S-AS=I-A^{n+1}</math>
:<math>S=(I-A)^{-1}(I-A^{n+1})</math>
 
If I-A is nonsingular one can evaluate the expression for the sum S.
 
==See also==
 
*[[Matrix exponential]]
 
{{DEFAULTSORT:Matrix Polynomial}}
[[Category:Matrices]]
[[Category:Polynomials]]
 
 
{{algebra-stub}}

Latest revision as of 20:27, 9 January 2015

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