Matrices with Polynomial Entries

Abstract

A polynomial in one variable is an expression of the form a _n x ⁿ + a _n−1 x ⁿ⁻¹ + ⋯ +a ₀, where x is a variable*, where n is a nonnegative integer, and where a _n , a _n-1, ⋯ , a ₀ are numbers. (Here, and in the remainder of the book, “number” will mean “rational number” unless otherwise stated.) The a’s are called the coefficients of the polynomial. The leading term of a polynomial is the term a _i x ⁱ with a _i ≠ 0 for which i is as large as possible. Normally one assumes a _n ≠ 0, so a _n x ⁿ is the leading term. The leading coefficient of a polynomial is the coefficient of the leading term; the degree is the exponent of x in the leading term. A polynomial of degree 0 is simply a nonzero rational number. The polynomial 0 has no leading term; it is considered to have degree −∞ in order to make the degree of the product of two polynomials equal to the sum of their degrees in all cases. For a similar reason, the leading coefficient of the polynomial 0 is considered to be 0. A polynomial is called monic if its leading coefficient is 1.