Here you will learn what is the matrix polynomial definition with examples.
Let’s begin –
Matrix Polynomial
Definition : Let f(x) = \(a_0x^n\) + \(a_1x^{n – 1}\) + \(a_2x^{n – 2}\) + ……. + \(a_{n – 1}x\) + \(a_n\) be a polynomial and let A be a square matrix of order n. Then,
f(A) = \(a_0A^n\) + \(a_1A^{n – 1}\) + \(a_2A^{n – 2}\) + …. \(a_{n – 1}A\) + \(a_nI^n\)
is called a matrix polynomial.
For example, if f(x) = \(x^2\) – 3x + 2 is a polynomial and A is a square matrix, then f(A) = \(A^2\) – 3A + 2I is a matrix polynomial.
Also Read : Different Types of Matrices – Definitions and Examples
Example : Let f(x) = \(x^2\) – 4x + 7. Find f(A), if A = \(\begin{bmatrix} 2 & 3 \\ -1 & 2 \end{bmatrix}\).
Solution : We have, f(x) = \(x^2\) – 4x + 7
\(\therefore\) f(A) = \(A^2 – 4A + 7I_2\)
Now, \(A^2\) = \(\begin{bmatrix} 2 & 3 \\ -1 & 2 \end{bmatrix}\)\(\begin{bmatrix} 2 & 3 \\ -1 & 2 \end{bmatrix}\)
= \(\begin{bmatrix} 4 – 3 & 6 + 6 \\ -2 – 2 & -3 + 4 \end{bmatrix}\) = \(\begin{bmatrix} 1 & 12 \\ -4 & 1 \end{bmatrix}\)
-4A = \(\begin{bmatrix} -8 & -12 \\ 4 & -8 \end{bmatrix}\)
and, \(7I_2\) = \(\begin{bmatrix} 7 & 0 \\ 0 & 7 \end{bmatrix}\)
\(\therefore\) f(A) = \(A^2 – 4A + 7I_2\)
\(\implies\) f(A) = \(\begin{bmatrix} 1 & 12 \\ -4 & 1 \end{bmatrix}\) + \(\begin{bmatrix} -8 & -12 \\ 4 & -8 \end{bmatrix}\) + \(\begin{bmatrix} 7 & 0 \\ 0 & 7 \end{bmatrix}\)
\(\implies\) f(A) = \(\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}\)
