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Chapter 9: Problem 32

Find a matrix \(A\) whose minimal polynomial is \(f(t)=t^{3}-8 t^{2}+5 t+7\). Simply let $A=\left[\begin{array}{lll}0 & 0 & -7 \\ 1 & 0 & -5 \\ 0 & 1 & 8\end{array}\right]\(, the companion matrix of \)f(t)$ [defined in Example 9.12(b)].

Short Answer

Expert verified
The matrix A with the given minimal polynomial \(f(t) = t^3 - 8t^2 + 5t + 7\) is its companion matrix, which is: \(A = \begin{bmatrix} 0 & 0 & -7 \\ 1 & 0 & -5 \\ 0 & 1 & 8 \end{bmatrix}\)

Step by step solution

01

Write down the minimal polynomial

We are given the minimal polynomial as \(f(t) = t^3 - 8t^2 + 5t + 7\).
02

Create the companion matrix

To create the companion matrix of the given polynomial, we follow these guidelines: 1. Start with a matrix of size nxn, where n is the degree of the polynomial, which in our case is 3. 2. Fill the diagonal below the main diagonal with 1's. 3. Place the constants of each power of t in the last row in reverse order, ending with the constant term, and negate them because they need to be subtracted. Using these guidelines, we create the companion matrix A as: \(A = \begin{bmatrix} 0 & 0 & -7 \\ 1 & 0 & -5 \\ 0 & 1 & 8 \end{bmatrix}\) The companion matrix A represents a matrix whose minimal polynomial is given by \(f(t) = t^3 - 8t^2 + 5t + 7\).

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Most popular questions from this chapter

Prove the following for a linear operator (matrix) \(T\) (a) The scalar 0 is an eigenvalue of \(T\) if and only if \(T\) is singular. (b) If \(\lambda\) is an eigenvalue of \(T,\) where \(T\) is invertible, then \(\lambda^{-1}\) is an eigenvalue of \(T^{-1}\)

Prove the Cayley-Hamilton Theorem 9.2: Every matrix \(A\) is a root of its characterstic polynomial \(\Delta(t)\)

Let \(f(t)\) and \(g(t)\) be monic polynomials (leading coefficient one) of minimal degree for which \(A\) is a root. Show \(f(t)=g(t) .\) [Thus, the minimal polynomial of \(A\) is unique.

Suppose \(v\) is an eigenvector of a linear operator \(T\) belonging to the eigenvalue \(\lambda\). Prove (a) For \(n>0, v\) is an cigcnvector of \(T^{n}\) belonging to \(\lambda^{n}\) (b) \(f(\lambda)\) is an cigenvalue of \(f(T)\) for any polynomial \(f(t)\)

Prove Theorem \(9.10^{\prime}:\) The geometric multiplicity of an eigenvalue \(\lambda\) of \(T\) does not exceed its algebraic multiplicity.
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