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

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

this polynomial evaluated at a matrix ${\displaystyle A}$ is

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

where ${\displaystyle I}$ is the identity matrix.^[1]

Note that ${\displaystyle P(A)}$ has the same dimension as ${\displaystyle A}$.

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

There is a unique monic polynomial of minimal degree which annihilates ${\displaystyle A}$ ; this polynomial is the minimal polynomial. Any polynomial which annihilates ${\displaystyle A}$  (such as the characteristic polynomial) is a multiple of the minimal polynomial.^[2]

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

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

where ${\displaystyle P^{(j)}}$  denotes the ${\displaystyle j}$ th derivative of ${\displaystyle P}$  and ${\displaystyle \lambda _{1},\dots ,\lambda _{s}}$  are the eigenvalues of ${\displaystyle A}$  with corresponding indices ${\displaystyle n_{1},\dots ,n_{s}}$  (the index of an eigenvalue is the size of its largest Jordan block).^[3]

Matrix geometrical series

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

${\displaystyle S=I+A+A^{2}+\cdots +A^{n}}$ 

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

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

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

If ${\displaystyle I-A}$  is nonsingular one can evaluate the expression for the sum ${\displaystyle S}$ .

See also

• Latimer–MacDuffee theorem
• Matrix exponential
• Matrix function

Notes

1. Horn & Johnson 1990, p. 36.
2. Horn & Johnson 1990, Thm 3.3.1.
3. Higham 2000, Thm 1.3.

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..