A matrix polynomial is an algebraic expression in which a polynomial’s indeterminate is replaced by a square matrix, or where the coefficients of the polynomial are themselves matrices. Formally, given a polynomial
$$ p(\lambda)=a_0 + a_1 \lambda + a_2 \lambda^2 + \dots + a_k \lambda^k, $$
with scalar coefficients $a_i \in \mathbb{F}$ (where $\mathbb{F}$ is a field such as $\mathbb{R}$ or $\mathbb{C}$), and a square matrix $A \in \mathbb{F}^{n\times n}$, the matrix polynomial $p(A)$ is defined by
$$ p(A)=a_0 I_n + a_1 A + a_2 A^2 + \dots + a_k A^k, $$
where $I_n$ denotes the $n\times n$ identity matrix and powers of $A$ are computed by matrix multiplication.
The notion can be extended to polynomials whose coefficients are matrices of compatible dimensions. If $B_0, B_1,\dots , B_k$ are matrices such that each product $B_i A^i$ is defined, the matrix polynomial
$$ q(A)=B_0 + B_1 A + B_2 A^2 + \dots + B_k A^k $$
is likewise well defined.
Key Properties
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Commutativity with the matrix argument: Since matrix multiplication is generally non‑commutative, the order of factors matters when coefficients are matrices. In the scalar‑coefficient case, each term $a_i A^i$ commutes with the identity scalar, but the powers $A^i$ themselves need not commute with other matrices.
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Evaluation consistency: If two polynomials $p$ and $q$ are identical as scalar functions (i.e., $p(\lambda)=q(\lambda)$ for all $\lambda$), then $p(A)=q(A)$ for any matrix $A$ that satisfies the polynomial identity, particularly when $A$ is diagonalizable.
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Cayley–Hamilton theorem: Every square matrix $A$ satisfies its own characteristic polynomial $\chi_A(\lambda)=\det(\lambda I_n - A)$; i.e., $\chi_A(A)=0_{n\times n}$. This theorem provides a fundamental example of a matrix polynomial that yields the zero matrix.
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Minimal polynomial: The minimal polynomial $m_A(\lambda)$ of a matrix $A$ is the monic polynomial of least degree such that $m_A(A)=0$. It divides any other polynomial that annihilates $A$.
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Spectral mapping theorem: For a matrix $A$ with spectrum $\sigma(A)$ (the set of eigenvalues), the spectrum of $p(A)$ satisfies
$$ \sigma(p(A)) = { p(\lambda) : \lambda \in \sigma(A) }. $$
Thus eigenvalues of a matrix polynomial are obtained by applying the scalar polynomial to the eigenvalues of the original matrix.
Applications
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Control theory: Transfer functions of linear time‑invariant systems are often expressed as rational matrix functions, which involve matrix polynomials in numerator and denominator.
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Differential equations: Solutions to linear differential equations with constant coefficients can be expressed using matrix exponentials, which are limits of matrix polynomials derived from the power series of the exponential function.
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Numerical linear algebra: Iterative methods (e.g., the Lanczos and Arnoldi processes) construct polynomial approximations to functions of matrices, such as the matrix sign function or the square root.
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Quantum mechanics and physics: Hamiltonians expressed as polynomials in operators are treated as matrix polynomials after discretization or when represented in a finite basis.
Computational Considerations
Evaluating a matrix polynomial directly by forming each power $A^i$ can be computationally intensive and numerically unstable. Techniques such as Horner’s method for matrices, Paterson–Stockmeyer algorithm, or exploiting sparsity and structure (e.g., banded or symmetric matrices) are employed to reduce the number of matrix multiplications and improve accuracy.
Related Concepts
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Polynomial matrix: A matrix whose entries are scalar polynomials in an indeterminate $\lambda$. This is distinct from a matrix polynomial, where the indeterminate is replaced by a matrix.
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Rational matrix function: A quotient of two matrix polynomials, often appearing in system theory and model reduction.
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Matrix function: A broader class encompassing analytic functions of matrices (e.g., exponential, logarithm), of which matrix polynomials are a special case obtained from truncating power series.
