Matrix polynomial

Not to be confused with Polynomial matrix.

In mathematics, a matrix polynomial is a polynomial with square matrices as variables. Given an ordinary, scalar-valued polynomial

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

this polynomial evaluated at a matrix $A$ is

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

where $I$ is the identity matrix.

Note that $P(A)$ has the same dimension as $A$.

A matrix polynomial equation is an equality between two matrix polynomials, which holds for the specific matrices in question. A matrix polynomial identity is a matrix polynomial equation which holds for all matrices A in a specified matrix ring M_n(R).

Matrix polynomials are often demonstrated in undergraduate linear algebra classes due to their relevance in showcasing properties of linear transformations represented as matrices, most notably the Cayley-Hamilton theorem.

Characteristic and minimal polynomial

The characteristic polynomial of a matrix A is a scalar-valued polynomial, defined by $p_{A}(t)=\det \left(tI-A\right)$. The Cayley–Hamilton theorem states that if this polynomial is viewed as a matrix polynomial and evaluated at the matrix $A$ itself, the result is the zero matrix: $p_{A}(A)=0$. An polynomial annihilates $A$ if $p(A)=0$; $p$ is also known as an annihilating polynomial. Thus, the characteristic polynomial is a polynomial which annihilates $A$.

There is a unique monic polynomial of minimal degree which annihilates $A$; this polynomial is the minimal polynomial. Any polynomial which annihilates $A$ (such as the characteristic polynomial) is a multiple of the minimal polynomial.

It follows that given two polynomials $P$ and $Q$, we have $P(A)=Q(A)$ if and only if

$P^{(j)}(\lambda _{i})=Q^{(j)}(\lambda _{i})\qquad {\text{for }}j=0,\ldots ,n_{i}-1{\text{ and }}i=1,\ldots ,s,$

where $P^{(j)}$ denotes the $j$th derivative of $P$ and $\lambda _{1},\dots ,\lambda _{s}$ are the eigenvalues of $A$ with corresponding indices $n_{1},\dots ,n_{s}$ (the index of an eigenvalue is the size of its largest Jordan block).

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

• Latimer–MacDuffee theorem
• Matrix exponential
• Matrix function

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 978-0-898716-81-8. Zbl 1170.15300.
• Higham, Nicholas J. (2000). Functions of Matrices: Theory and Computation. SIAM. ISBN 089-871-777-9..
• Horn, Roger A.; Johnson, Charles R. (1990). Matrix Analysis. Cambridge University Press. ISBN 978-0-521-38632-6..