Matrix polynomial

Not to be confused with Polynomial matrix.

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.

$\left[A,B\right]=AB-BA,$

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 equation $P(\lambda )=Q(\lambda )$.
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

References

• Gohberg, Israel; Lancaster, Peter; Rodman, Leiba (2009) [1982]. Matrix Polynomials. Classics in Applied Mathematics. Vol. 58. Lancaster, PA: Society for Industrial and Applied Mathematics. ISBN 0-898716-81-0. Zbl 1170.15300.