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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

Show that \(I - A\) is invertible when all the eigenvalues of \(A\) are less than 1 in magnitude. (Hint: What would be true if \(I - A\) were not invertible?)

It is proved that \(I - A\) is invertible when all the eigenvalues of \(A\) are less than 1 in magnitude.

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Definition of magnitude

The eigenvalue \(\lambda \) is the real or complex number of a matrix \(A\) which is a square matrix that satisfies the following equation

\(\det \left( {A - \lambda I} \right) = 0\).

This equation is called the characteristic equation.

Step 2: Showing that \(I - A\) is invertible

If \(I - A\) is not invertible, then the equation\(\left( {I - A} \right){\rm{x}} = 0\) would have a non-trivial solution of \({\rm{x}}\).

Then \({\rm{x}} - A{\rm{x}} = 0\) and \(A{\rm{x}} = 1 \cdot {\rm{x}}\), which shows that \(A\) would have \(1\) as an

eigenvalue.

This cannot happen if all the eigenvalues are less than \(1\)in magnitude.

So \(I - A\) must be invertible.

It is proved that \(I - A\) is invertible when all the eigenvalues of \(A\) are less than 1 in magnitude.

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 \(A\).

[M] Repeat Exercise 25 for \[A{\bf{ = }}\left[ {\begin{array}{*{20}{c}}{{\bf{ - 8}}}&{\bf{5}}&{{\bf{ - 2}}}&{\bf{0}}\\{{\bf{ - 5}}}&{\bf{2}}&{\bf{1}}&{{\bf{ - 2}}}\\{{\bf{10}}}&{{\bf{ - 8}}}&{\bf{6}}&{{\bf{ - 3}}}\\{\bf{3}}&{{\bf{ - 2}}}&{\bf{1}}&{\bf{0}}\end{array}} \right]\].

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

3. \(\left[ {\begin{array}{*{20}{c}}3&-2\\1&-1\end{array}} \right]\)

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.

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

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

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