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Chapter 5: Q5.3-22E (page 267)

Question: In Exercises 21 and 22, A, B, Pand Dare \(n \times n\) matrices. Mark each statement True or False. Justify each answer.(Study Theorem 5 and 6 and the examples in this section carefully before you try these exercises.)

22.

a. A is diagonalizable if A has neigenvectos.

b. If A is diagonizable, then A has n eigenvalues.

c. If\(AP = PD\), with D diagonal, then the nonzero columns of P must be eigenvectors of A.

d. If A is invertible, then A is diagonizable.

Short Answer

Expert verified

a. The given statement is False.

b. The given statement is False.

c. The given statement is True.

d. The given statement is False.

Step by step solution

01

Find an answer for part (a)

According to the diagonalization theorem, the n eigenvectors must be linearly independent.

Therefore, the given statement is False.

02

Find an answer for part (b)

According to theorem 6, If a \(n \times n\) matrix with \(n\) distinct eigenvalues is diagonalizable.

The given is the converse of Theorem 6.

Therefore, the given statement is False.

03

Find an answer for part (c)

According to a diagonalizable theorem, if\(AP = PD\)then,\(A = PD{P^{ - 1}}\)and D must be a diagonal matrix.The previous statement will be true if columns of P are linearly independent eigenvectors of matrix A.

The diagonal elements of D represent the eigenvalues of A.

Therefore, the given statement is True.

04

Find an answer for part (d)

For a matrix to be invertible, its columns must be linearly independent and for a matrix to be diagonalizable, it must have n linearly independent vectors.

Therefore, the given statement is False.

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Most popular questions from this chapter

Exercises 19–23 concern the polynomial \(p\left( t \right) = {a_{\bf{0}}} + {a_{\bf{1}}}t + ... + {a_{n - {\bf{1}}}}{t^{n - {\bf{1}}}} + {t^n}\) and \(n \times n\) matrix \({C_p}\) called the companion matrix of \(p\): \({C_p} = \left( {\begin{aligned}{*{20}{c}}{\bf{0}}&{\bf{1}}&{\bf{0}}&{...}&{\bf{0}}\\{\bf{0}}&{\bf{0}}&{\bf{1}}&{}&{\bf{0}}\\:&{}&{}&{}&:\\{\bf{0}}&{\bf{0}}&{\bf{0}}&{}&{\bf{1}}\\{ - {a_{\bf{0}}}}&{ - {a_{\bf{1}}}}&{ - {a_{\bf{2}}}}&{...}&{ - {a_{n - {\bf{1}}}}}\end{aligned}} \right)\).

19. Write the companion matrix \({C_p}\) for \(p\left( t \right) = {\bf{6}} - {\bf{5}}t + {t^{\bf{2}}}\), and then find the characteristic polynomial of \({C_p}\).

Question: Find the characteristic polynomial and the eigenvalues of the matrices in Exercises 1-8.

4. \(\left[ {\begin{array}{*{20}{c}}5&-3\\-4&3\end{array}} \right]\)

Define\(T:{{\rm P}_3} \to {\mathbb{R}^4}\)by\(T\left( {\bf{p}} \right) = \left( {\begin{aligned}{{\bf{p}}\left( { - 3} \right)}\\{{\bf{p}}\left( { - 1} \right)}\\{{\bf{p}}\left( 1 \right)}\\{{\bf{p}}\left( 3 \right)}\end{aligned}} \right)\).

  1. Show that \(T\) is a linear transformation.
  2. Find the matrix for \(T\) relative to the basis \(\left\{ {1,t,{t^2},{t^3}} \right\}\)for \({{\rm P}_3}\)and the standard basis for \({\mathbb{R}^4}\).

Exercises 19–23 concern the polynomial \(p\left( t \right) = {a_{\bf{0}}} + {a_{\bf{1}}}t + ... + {a_{n - {\bf{1}}}}{t^{n - {\bf{1}}}} + {t^n}\) and \(n \times n\) matrix \({C_p}\) called the companion matrix of \(p\): \({C_p} = \left( {\begin{aligned}{*{20}{c}}{\bf{0}}&{\bf{1}}&{\bf{0}}&{...}&{\bf{0}}\\{\bf{0}}&{\bf{0}}&{\bf{1}}&{}&{\bf{0}}\\:&{}&{}&{}&:\\{\bf{0}}&{\bf{0}}&{\bf{0}}&{}&{\bf{1}}\\{ - {a_{\bf{0}}}}&{ - {a_{\bf{1}}}}&{ - {a_{\bf{2}}}}&{...}&{ - {a_{n - {\bf{1}}}}}\end{aligned}} \right)\).

20. Let \(p\left( t \right){\bf{ = }}\left( {t{\bf{ - 2}}} \right)\left( {t{\bf{ - 3}}} \right)\left( {t{\bf{ - 4}}} \right){\bf{ = - 24 + 26}}t{\bf{ - 9}}{t^{\bf{2}}}{\bf{ + }}{t^{\bf{3}}}\). Write the companion matrix for \(p\left( t \right)\), and use techniques from chapter \({\bf{3}}\) to find the characteristic polynomial.

For the matrices Afind real closed formulas for the trajectoryx→(t+1)=Ax→(t)wherex→(0)=[01]. Draw a rough sketchA=[1-31.2-2.6]
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