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Chapter 4: Problem 12

Let \(A\) and \(B\) be similar matrices. Show that (a) \(A^{T}\) and \(B^{T}\) are similar. (b) \(A^{k}\) and \(B^{k}\) are similar for each positive integer \(k\)

Short Answer

Expert verified
(a) Since A and B are similar, we have A = PBP^{-1} for some invertible matrix P. Taking the transpose yields A^{T} = (PBP^{-1})^T = (P^{-1})^TB^TP^T = (P^T)^{-1}B^TP^T, which shows that A^{T} and B^{T} are similar. (b) We are given A = PBP^{-1}, so raising both sides to the k-th power and using the properties of the identity matrix, we find A^{k} = PB^kP^{-1}. Thus, A^{k} and B^{k} are similar for each positive integer k.

Step by step solution

01

(a) Proving that A^{T} and B^{T} are similar

If A and B are similar, then we have A = PBP^{-1} for some invertible matrix P. The idea now is to find a relationship between A^{T} and B^{T}, using the given relationship between A and B. First, let's take the transpose of both sides of the equation A = PBP^{-1}: \( A^{T} = (PBP^{-1})^T \) Now, we can use the transpose properties: 1. (AB)^T = B^TA^T 2. (A^{-1})^T = (A^T)^{-1} Applying property (1) to the equation: \(A^T = (P^{-1})^TB^TP^T\) Now, let's apply property (2) to the P^{-1} term: \(A^T = (P^T)^{-1}B^TP^T \) This expression is of the form Q^{-1}B^TQ, where Q = P^T. This means A^{T} and B^{T} are similar matrices.
02

(b) Proving that A^{k} and B^{k} are similar for each positive integer k

We are given A = PBP^{-1}. We want to find A^{k} and B^{k} and check if they are also similar. Let's start by computing A^{k}: \( A^k = (PBP^{-1})(PBP^{-1})... (PBP^{-1}) \) (k times) Notice that the product of the form P^{-1}P appears k - 1 times in the expression: \( A^k = P(BP^{-1}PB) (BP^{-1}PB) ... (BP^{-1}PB)P^{-1} \) Since P^{-1}P is the identity matrix I, the expression simplifies: \( A^k = PB^kP^{-1} \) We have found that A^{k} can be written in terms of B^{k} and P. Thus, A^{k} and B^{k} are similar for each positive integer k.

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Most popular questions from this chapter

Let \(L\) be the linear operator on \(\mathbb{R}^{3}\) defined by \\[ L(\mathbf{x})=\left(\begin{array}{l} 2 x_{1}-x_{2}-x_{3} \\ 2 x_{2}-x_{1}-x_{3} \\ 2 x_{3}-x_{1}-x_{2} \end{array}\right) \\] Determine the standard matrix representation \(A\) of \(L,\) and use \(A\) to find \(L(\mathbf{x})\) for each of the following vectors x: (a) \(\mathbf{x}=(1,1,1)^{T}\) (b) \(\mathbf{x}=(2,1,1)^{T}\) (c) \(x=(-5,3,2)^{T}\)

Let \(\left\\{\mathbf{u}_{1}, \mathbf{u}_{2}\right\\}\) and \(\left\\{\mathbf{v}_{1}, \mathbf{v}_{2}\right\\}\) be ordered bases for \(\mathbb{R}^{2}\) where $$\mathbf{u}_{1}=\left(\begin{array}{l} 1 \\ 1 \end{array}\right), \quad \mathbf{u}_{2}=\left(\begin{array}{r} -1 \\ 1 \end{array}\right)$$ and $$\mathbf{v}_{1}=\left(\begin{array}{l} 2 \\ 1 \end{array}\right), \quad \mathbf{v}_{2}=\left(\begin{array}{l} 1 \\ 0 \end{array}\right)$$ Let \(L\) be the linear transformation defined by $$L(\mathbf{x})=\left(-x_{1}, x_{2}\right)^{T}$$ and let \(B\) be the matrix representing \(L\) with respect (a) Find the transition matrix \(S\) corresponding to the change of basis from \(\left\\{\mathbf{u}_{1}, \mathbf{u}_{2}\right\\}\) to \(\left\\{\mathbf{v}_{1}, \mathbf{v}_{2}\right\\}\) (b) Find the matrix \(A\) representing \(L\) with respect to \(\left\\{\mathbf{v}_{1}, \mathbf{v}_{2}\right\\}\) by computing \(S B S^{-1}\) (c) Verify that $$\begin{array}{l} L\left(\mathbf{v}_{1}\right)=a_{11} \mathbf{v}_{1}+a_{21} \mathbf{v}_{2} \\ L\left(\mathbf{v}_{2}\right)=a_{12} \mathbf{v}_{1}+a_{22} \mathbf{v}_{2} \end{array}$$

Use mathematical induction to prove that if \(L\) is a linear transformation from \(V\) to \(W\), then \\[ \begin{array}{l} L\left(\alpha_{1} \mathbf{v}_{1}+\alpha_{2} \mathbf{v}_{2}+\cdots+\alpha_{n} \mathbf{v}_{n}\right) \\ \quad=\alpha_{1} L\left(\mathbf{v}_{1}\right)+\alpha_{2} L\left(\mathbf{v}_{2}\right)+\cdots+\alpha_{n} L\left(\mathbf{v}_{n}\right) \end{array} \\]

Let \(\mathbf{y}_{1}, \mathbf{y}_{2},\) and \(\mathbf{y}_{3}\) be defined as in Exercise \(7,\) and let \(L\) be the linear operator on \(\mathbb{R}^{3}\) defined by \(L\left(c_{1} \mathbf{y}_{1}+c_{2} \mathbf{y}_{2}+c_{3} \mathbf{y}_{3}\right)\) \(=\left(c_{1}+c_{2}+c_{3}\right) \mathbf{y}_{1}+\left(2 c_{1}+c_{3}\right) \mathbf{y}_{2}-\left(2 c_{2}+c_{3}\right) \mathbf{y}_{3}\) (a) Find a matrix representing \(L\) with respect to the ordered basis \(\left\\{\mathbf{y}_{1}, \mathbf{y}_{2}, \mathbf{y}_{3}\right\\}\) (b) For each of the following, write the vector \(\mathbf{x}\) as a linear combination of \(\mathbf{y}_{1}, \mathbf{y}_{2},\) and \(\mathbf{y}_{3}\) and use the matrix from part (a) to determine \(L(\mathbf{x})\) (i) \(\mathbf{x}=(7,5,2)^{T}\) (ii) \(\quad \mathbf{x}=(3,2,1)^{T}\) (iii) \(\mathbf{x}=(1,2,3)^{T}\)

Suppose that \(L_{1}: V \rightarrow W\) and \(L_{2}: W \rightarrow Z\) are linear transformations and \(E, F,\) and \(G\) are ordered bases for \(V, W,\) and \(Z,\) respectively. Show that, if \(A\) represents \(L_{1}\) relative to \(E\) and \(F\) and \(B\) represents \(L_{2}\) relative to \(F\) and \(G,\) then the matrix \(C=B A\) represents \(L_{2} \circ L_{1}: V \rightarrow Z\) relative to \(E\) and \(G .\left[\text { Hint: Show that } B A[\mathbf{v}]_{E}=\left[\left(L_{2} \circ L_{1}\right)(\mathbf{v})\right]_{G}\right.\) for all \(\mathbf{v} \in V .]\)
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