Matrix polynomials. Let p(x)=cn x^n+cn-1 x^n-1+⋯+c1 x+c0 be a polynomial function. If A is a squ

step 1 We want to prove that px equals to equal to p1 x p2 x while a equal to 3 1 2 1 2 1 x2 minus 9 an

Matrix polynomials. Let p(x)=cn x^n+cn-1 x^n-1+⋯+c1 x+c0 be...

Matrix polynomials. Let $p(x)=c_n x^n+c_{n-1} x^{n-1}+\cdots+c_1 x+c_0$ be a polynomial function. If $A$ is a square matrix, we define the corresponding matrix polynomial $p(A)=$ $c_n A^n+c_{n-1} A^{n-1}+\cdots+c_1 A+c_0 I$; the constant term becomes a scalar multiple of the identity matrix. For instance, if $p(x)=x^2-2 x+3$, then $p(A)=A^2-2 A+3 \mathrm{I}$. (a) Write out the matrix polynomials $p(A), q(A)$ when $p(x)=x^3-3 x+2, q(x)=2 x^2+1$. (b) Evaluate $p(A)$ and $q(A)$ when $A=\left(\begin{array}{rr}1 & 2 \\ -1 & -1\end{array}\right)$. (c) Show that the matrix product $p(A) q(A)$ is the matrix polynomial corresponding to the product polynomial $r(x)=p(x) q(x)$. (d) True or false: If $B=p(A)$ and $C=q(A)$, then $B C=C B$. Check your answer in the particular case of part (b).

Key Concepts

Matrix Polynomials

A matrix polynomial is a polynomial in which the variable is replaced by a square matrix. Given a polynomial p(x) = c?x? + c???x??¹ + … + c?x + c?, the corresponding matrix polynomial is defined as p(A) = c?A? + c???A??¹ + … + c?A + c?I, where A is a square matrix, A? denotes the matrix multiplied by itself k times, and I is the identity matrix. This definition ensures that the constant term, which is a scalar in the original polynomial, becomes a scalar multiple of the identity matrix when evaluated at a matrix.

Substitution of Matrices into Polynomials

The process of replacing the variable x in a polynomial with a square matrix is known as evaluating the polynomial at that matrix. This substitution respects the operations of scalar multiplication, matrix addition, and matrix multiplication, allowing one to compute polynomial expressions where each term is a matrix power multiplied by the corresponding polynomial coefficient. It is an important concept because it extends polynomial functions from numbers to matrices while retaining much of their algebraic structure.

Identity Matrix in Matrix Polynomials

When evaluating a polynomial at a matrix, the role of the constant term is taken over by the identity matrix scaled by that constant. This is crucial because the identity matrix acts as the multiplicative identity in matrix algebra (analogous to the number 1 in scalar algebra), ensuring that the constant term is correctly represented in the polynomial expression. It maintains the consistency and validity of the polynomial function when transitioning from scalar variables to matrices.

Polynomial Multiplication and Matrix Product

One key property of matrix polynomials is that the multiplication of two matrix polynomials corresponds to evaluating the product of their underlying scalar polynomials. Specifically, if p(x) and q(x) are polynomials and p(A) and q(A) are their corresponding matrix polynomials, then p(A)q(A) is equal to r(A), where r(x) is the product polynomial p(x)q(x). This property illustrates the homomorphic nature of the polynomial map and is fundamental in connecting polynomial algebra with matrix algebra.

Noncommutativity of Matrix Multiplication

Unlike scalar multiplication, matrix multiplication is generally not commutative; that is, for matrices B and C, it is often the case that BC ? CB. Even when matrices are functions of another matrix (for example, B = p(A) and C = q(A)), one might initially expect them to commute because they are both derived from the same matrix A. However, while polynomial functions of a matrix may commute under certain conditions, the general noncommutative nature of matrix multiplication requires careful verification of such properties in any given case.

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