Matrix polynomial

Short description: Material matrix

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

[math]\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, }[/math]

this polynomial evaluated at a matrix [math]\displaystyle{ A }[/math] is

[math]\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, }[/math]

where [math]\displaystyle{ I }[/math] is the identity matrix.

Note that [math]\displaystyle{ P(A) }[/math] has the same dimension as [math]\displaystyle{ A }[/math].

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

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

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

[math]\displaystyle{ P^{(j)}(\lambda_i) = Q^{(j)}(\lambda_i) \qquad \text{for } j = 0,\ldots,n_i-1 \text{ and } i = 1,\ldots,s, }[/math]

where [math]\displaystyle{ P^{(j)} }[/math] denotes the [math]\displaystyle{ j }[/math]th derivative of [math]\displaystyle{ P }[/math] and [math]\displaystyle{ \lambda_1, \dots, \lambda_s }[/math] are the eigenvalues of [math]\displaystyle{ A }[/math] with corresponding indices [math]\displaystyle{ n_1, \dots, n_s }[/math] (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,

[math]\displaystyle{ S=I+A+A^2+\cdots +A^n }[/math]

[math]\displaystyle{ AS=A+A^2+A^3+\cdots +A^{n+1} }[/math]

[math]\displaystyle{ (I-A)S=S-AS=I-A^{n+1} }[/math]

[math]\displaystyle{ S=(I-A)^{-1}(I-A^{n+1}) }[/math]

If [math]\displaystyle{ I - A }[/math] is nonsingular one can evaluate the expression for the sum [math]\displaystyle{ S }[/math].

See also

• Latimer–MacDuffee theorem
• Matrix exponential
• Matrix function

References

• Gohberg, Israel; Lancaster, Peter; Rodman, Leiba (2009). Matrix Polynomials. Classics in Applied Mathematics. 58. Lancaster, PA: Society for Industrial and Applied Mathematics. ISBN 978-0-898716-81-8. 
• 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. .