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

is

is the identity matrix.

.

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 Mn(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.

The determinant of a matrix polynomial with Hermitian positive-definite (semidefinite) coefficients is a polynomial with positive (nonnegative) coefficients.

Characteristic and minimal polynomial

The characteristic polynomial of a matrix A. The Cayley–Hamilton theorem. A polynomial annihilates is also known as an annihilating polynomial.

There is a unique monic polynomial; this polynomial is the minimal polynomial (such as the characteristic polynomial) is a multiple of the minimal polynomial.

if and only if

are the eigenvalues (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,

.

See also

• Latimer–MacDuffee theorem

• Matrix exponential

• Matrix function

Notes

References

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