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Q19SE

Expert-verified

Linear Algebra and its Applications

Found in: Page 267

Book edition 5th

Author(s) David C. Lay, Steven R. Lay and Judi J. McDonald

Pages 483 pages

ISBN 978-03219822384

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Short Answer

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 \({C_p}\).

The characteristic polynomial of the matrix \({C_p}\) is \(p\left( \lambda \right)\).

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Compare the given companion matrix with the given polynomial.

Consider the polynomial \(p\left( t \right) = {a_0} + {a_1}t + ... + {a_{n - 1}}{t^{n - 1}} + {t^n}\).

The companion matrix of \(p\) is \({C_p} = \left( {\begin{aligned}{*{20}{c}}0&1&0&{...}&0\\0&0&1&{}&0\\:&{}&{}&{}&:\\0&0&0&{}&1\\{ - {a_0}}&{ - {a_1}}&{ - {a_2}}&{...}&{ - {a_{n - 1}}}\end{aligned}} \right)\).

Thus, we get,

\({C_p} = \left( {\begin{aligned}{*{20}{c}}0&1\\{ - 6}&5\end{aligned}} \right)\)

Step 2: Find the characteristic polynomial

\(\begin{aligned}{c}\det \left( {{C_p} - \lambda I} \right) &= \det \left( {\begin{aligned}{*{20}{c}}{0 - \lambda }&1\\{ - 6}&{5 - \lambda }\end{aligned}} \right)\\ &= \left( { - \lambda } \right)\left( {5 - \lambda } \right) + 6\\ &= - 5\lambda + {\lambda ^2} + 6\\ &= 6 - 5\lambda + {\lambda ^2}\\ &= p\left( \lambda \right)\end{aligned}\)

Therefore, the characteristic polynomial of the matrix \({C_p}\) is \(p\left( \lambda \right)\).

Most popular questions for Math Textbooks

Mark each statement as True or False. Justify each answer.

a. If \(A\) is invertible and 1 is an eigenvalue for \(A\), then \(1\) is also an eigenvalue of \({A^{ - 1}}\)

b. If \(A\) is row equivalent to the identity matrix \(I\), then \(A\) is diagonalizable.

c. If \(A\) contains a row or column of zeros, then 0 is an eigenvalue of \(A\)

d. Each eigenvalue of \(A\) is also an eigenvalue of \({A^2}\).

e. Each eigenvector of \(A\) is also an eigenvector of \({A^2}\)

f. Each eigenvector of an invertible matrix \(A\) is also an eigenvector of \({A^{ - 1}}\)

g. Eigenvalues must be nonzero scalars.

h. Eigenvectors must be nonzero vectors.

i. Two eigenvectors corresponding to the same eigenvalue are always linearly dependent.

j. Similar matrices always have exactly the same eigenvalues.

k. Similar matrices always have exactly the same eigenvectors.

I. The sum of two eigenvectors of a matrix \(A\) is also an eigenvector of \(A\).

m. The eigenvalues of an upper triangular matrix \(A\) are exactly the nonzero entries on the diagonal of \(A\).

n. The matrices \(A\) and \({A^T}\) have the same eigenvalues, counting multiplicities.

o. If a \(5 \times 5\) matrix \(A\) has fewer than 5 distinct eigenvalues, then \(A\) is not diagonalizable.

p. There exists a \(2 \times 2\) matrix that has no eigenvectors in \({A^2}\)

q. If \(A\) is diagonalizable, then the columns of \(A\) are linearly independent.

r. A nonzero vector cannot correspond to two different eigenvalues of \(A\).

s. A (square) matrix \(A\) is invertible if and only if there is a coordinate system in which the transformation \({\bf{x}} \mapsto A{\bf{x}}\) is represented by a diagonal matrix.

t. If each vector \({{\bf{e}}_j}\) in the standard basis for \({A^n}\) is an eigenvector of \(A\), then \(A\) is a diagonal matrix.

u. If \(A\) is similar to a diagonalizable matrix \(B\), then \(A\) is also diagonalizable.

v. If \(A\) and \(B\) are invertible \(n \times n\) matrices, then \(AB\)is similar to \ (BA\ )

w. An \(n \times n\) matrix with \(n\) linearly independent eigenvectors is invertible.

x. If \(A\) is an \(n \times n\) diagonalizable matrix, then each vector in \({A^n}\) can be written as a linear combination of eigenvectors of \(A\).

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 \(p\): \({C_p} = \left( {\begin{aligned}{*{20}{c}}{\bf{0}}&{\bf{1}}&{\bf{0}}&{...}&{\bf{0}}\\{\bf{0}}&{\bf{0}}&{\bf{1}}&{}&{\bf{0}}\\:&{}&{}&{}&:\\{\bf{0}}&{\bf{0}}&{\bf{0}}&{}&{\bf{1}}\\{ - {a_{\bf{0}}}}&{ - {a_{\bf{1}}}}&{ - {a_{\bf{2}}}}&{...}&{ - {a_{n - {\bf{1}}}}}\end{aligned}} \right)\).

21. Use mathematical induction to prove that for \(n \ge {\bf{2}}\),\(\begin{aligned}{c}det\left( {{C_p} - \lambda I} \right) = {\left( { - {\bf{1}}} \right)^n}\left( {{a_{\bf{0}}} + {a_{\bf{1}}}\lambda + ... + {a_{n - {\bf{1}}}}{\lambda ^{n - {\bf{1}}}} + {\lambda ^n}} \right)\\ = {\left( { - {\bf{1}}} \right)^n}p\left( \lambda \right)\end{aligned}\)

(Hint: Expanding by cofactors down the first column, show that \(det\left( {{C_p} - \lambda I} \right)\) has the form \(\left( { - \lambda B} \right) + {\left( { - {\bf{1}}} \right)^n}{a_{\bf{0}}}\) where \(B\) is a certain polynomial (by the induction assumption).)

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

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

Question: Is \(\left( {\begin{array}{*{20}{c}}1\\4\end{array}} \right)\) an eigenvalue of \(\left( {\begin{array}{*{20}{c}}{ - 3}&1\\{ - 3}&8\end{array}} \right)\)? If so, find the eigenvalue.

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.

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