Permutation polynomial

Template:Distinguish In mathematics, a matrix polynomial is a polynomial with matrices as variables. Examples include:

$P(A)=\sum _{i=0}^n{a}_i{A}^i=a_{0}I+a_{1}A+a_2{A}^2+\cdots +a_n{A}^n,$

where P is a polynomial,

$P(x)=\sum _{i=0}^n{a}_i{x}^i=a_0+a_{1}x+a_2{x}^2+\cdots +a_n{x}^n,$

and I is the identity matrix.

$[A,B]=AB-BA,$

the 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 $P(A)=Q(A)$, (where A is a matrix over a field), then the eigenvalues of A satisfy the characteristic equationTemplate:Disputed-inline $P(\lambda )=Q(\lambda )$.
A matrix polynomial identity is a matrix polynomial equation which holds for all matricies A in a specified matrix ring M_n(R).

Matrix geometrical series

Matrix polynomials can be used to sum a matrix geometrical series as one would an ordinary geometric series,

$S=I+A+A^2+\cdots +A^n$

$AS=A+A^2+A^3+\cdots +A^{n+1}$

$(I-A)S=S-AS=I-A^{n+1}$

$S=(I-A{)}^{-1}(I-A^{n+1})$

If I-A is nonsingular one can evaluate the expression for the sum S.

See also

• Matrix exponential

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