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Linear Algebra and its Applications
Found in: Page 267
Linear Algebra and its Applications

Linear Algebra and its Applications

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

(M) Use a matrix program to diagonalize

\(A = \left( {\begin{aligned}{*{20}{c}}{ - 3}&{ - 2}&0\\{14}&7&{ - 1}\\{ - 6}&{ - 3}&1\end{aligned}} \right)\)

If possible. Use the eigenvalue command to create the diagonal matrix \(D\). If the program has a command that produces eigenvectors, use it to create an invertible matrix \(P\). Then compute \(AP - PD\) and \(PD{P^{{\bf{ - 1}}}}\). Discuss your results.

\(A\) is not diagonalizable. Use command eig in MATLAB to get invertible matrix \(P\) and diagonal matrix \(D\) if possible.

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1: Determine whether \(P\) is invertible or not

First, we need to define a matrix \(A\) in MATLAB and use command eig to get an invertible matrix \(P\) and a diagonal matrix \(D\).

\(\left( {\begin{aligned}{*{20}{c}}P&D\end{aligned}} \right) = {\rm{eig}}\left( A \right)\)

Therefore,

\(P = \left( {\begin{aligned}{*{20}{c}}{0.3244}&{ - 0.3244}&{0.3333}\\{ - 0.8111}&{0.8111}&{ - 0.6667}\\{0.4867}&{ - 0.4867}&{0.6667}\end{aligned}} \right)\)

And

\(D = \left( {\begin{aligned}{*{20}{c}}2&0&0\\0&2&0\\0&0&1\end{aligned}} \right)\)

As the matrix \(A\) has two eigenvalues \(2\) and \(1\).

Since the first two columns of the matrix are linearly dependent that means the eigenvalue \(2\) is dimensional that shows \(P\) is not invertible.

Therefore \(A\) is not diagonalizable.

Step 2: Compute \(AP - PD\) and \(PD{P^{{\bf{ - 1}}}}\)

Since the matrix \(P\) is not invertible therefore \(PD{P^{ - 1}}\) cannot be computed.

However, \(AP - PD\) can be computed.

\(AP\)

\(PD\)

\(AP - PD\)

As \(AP - PD\) is equal to zero matrix.

Since \(P\) consists of eigenvectors of \(A\) therefore columns of \(AP - PD\) are eigenvectors of \(A\) which are multiplied by its eigenvalues.

Also, \(D\) consists of eigenvalues of \(A\) and columns of \(PD\) are eigenvectors of \(A\) multiplied by its eigenvalues.

Thus, use command eig in MATLAB to get invertible matrix \(P\) and diagonal matrix \(D\) if possible.

Most popular questions for Math Textbooks

Consider an invertible n × n matrix A such that the zero state is a stable equilibrium of the dynamical system x→(t+1)=Ax→(t) What can you say about the stability of the systems

x→(t+1)=A-1x→(t)

M] In Exercises 19 and 20, find (a) the largest eigenvalue and (b) the eigenvalue closest to zero. In each case, set \[{{\bf{x}}_{\bf{0}}}{\bf{ = }}\left( {{\bf{1,0,0,0}}} \right)\] and carry out approximations until the approximating sequence seems accurate to four decimal places. Include the approximate eigenvector.

19.\[A{\bf{=}}\left[{\begin{array}{*{20}{c}}{{\bf{10}}}&{\bf{7}}&{\bf{8}}&{\bf{7}}\\{\bf{7}}&{\bf{5}}&{\bf{6}}&{\bf{5}}\\{\bf{8}}&{\bf{6}}&{{\bf{10}}}&{\bf{9}}\\{\bf{7}}&{\bf{5}}&{\bf{9}}&{{\bf{10}}}\end{array}} \right]\]

Suppose \(A = PD{P^{ - 1}}\), where \(P\) is \(2 \times 2\) and \(D = \left( {\begin{array}{*{20}{l}}2&0\\0&7\end{array}} \right)\)

a. Let \(B = 5I - 3A + {A^2}\). Show that \(B\) is diagonalizable by finding a suitable factorization of \(B\).

b. Given \(p\left( t \right)\) and \(p\left( A \right)\) as in Exercise 5 , show that \(p\left( A \right)\) is diagonalizable.

Question: In Exercises \({\bf{1}}\) and \({\bf{2}}\), let \(A = PD{P^{ - {\bf{1}}}}\) and compute \({A^{\bf{4}}}\).

2. \(P{\bf{ = }}\left( {\begin{array}{*{20}{c}}2&{ - 3}\\{ - 3}&5\end{array}} \right)\), \(D{\bf{ = }}\left( {\begin{array}{*{20}{c}}{\bf{1}}&{\bf{0}}\\{\bf{0}}&{\frac{{\bf{1}}}{{\bf{2}}}}\end{array}} \right)\)

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

22. Let \(p\left( t \right) = {a_{\bf{0}}} + {a_{\bf{1}}}t + {a_{\bf{2}}}{t^{\bf{2}}} + {t^{\bf{3}}}\), and let \(\lambda \) be a zero of \(p\).

  1. Write the companion matrix for \(p\).
  2. Explain why \({\lambda ^{\bf{3}}} = - {a_{\bf{0}}} - {a_{\bf{1}}}\lambda - {a_{\bf{2}}}{\lambda ^{\bf{2}}}\), and show that \(\left( {{\bf{1}},\lambda ,{\lambda ^2}} \right)\) is an eigenvector of the companion matrix for \(p\).
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