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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

If \(p\left( t \right) = {c_0} + {c_1}t + {c_2}{t^2} + ...... + {c_n}{t^n}\), define \(p\left( A \right)\) to be the matrix formed by replacing each power of \(t\) in \(p\left( t \right)\)by the corresponding power of \(A\) (with \({A^0} = I\) ). That is,

\(p\left( t \right) = {c_0} + {c_1}I + {c_2}{I^2} + ...... + {c_n}{I^n}\)

Show that if \(\lambda \) is an eigenvalue of A, then one eigenvalue of \(p\left( A \right)\) is \(p\left( \lambda \right)\).

It is proved that if \(\lambda \) is an eigenvalue of \(\lambda \), then one eigenvalue of \(p\left( A \right)\) is \(p\left( \lambda \right)\).

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Step by Step Solution

TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1: Definition of eigenvalue

Eigenvalues are a special set of scalars associated with a linear system of equations that are sometimes also known as characteristic roots, characteristic values, proper values, or latent roots.

Step 2: Showing if is an eigenvalue of \(A\), then one eigenvalue of \(p\left( A \right)\) is \(p\left( \lambda  \right)\)

Suppose that \(\lambda \) is an eigenvalue of \(A\) with the corresponding eigenvector\({\rm{x}}\), that is, suppose that \(A{\rm{x}} = \lambda {\rm{x}},{\rm{x}} \ne 0\).

Then,

\(\begin{array}{r}p\left( A \right){\rm{x}} = \left( {{c_0}I + {c_1}A + \ldots + {c_n}{A^n}} \right){\rm{x}}\\ = {c_0}{\rm{x}} + {c_1}A{\rm{x}} + \ldots + {c_n}{A^n}{\rm{x}}\end{array}\)

From exercise 4 we know that\({A^n}{\rm{x}} = {\lambda ^n}{\rm{x}}\), so:

\(\begin{array}{c}p\left( A \right){\rm{x}} = {c_0}{\rm{x}} + {c_1}\lambda {\rm{x}} + \ldots + {c_n}{\lambda ^n}{\rm{x}}\\ = \left( {{c_0} + {c_1}\lambda + \ldots + {c_n}{\lambda ^n}} \right){\rm{x}}\\ = p(\lambda ){\rm{x}}\end{array}\).

Hence it is proved that \(\lambda \) is an eigenvalue of \(\lambda \), then one eigenvalue of \(p\left( A \right)\) is \(p\left( \lambda \right)\).

Most popular questions for Math Textbooks

Suppose \({\bf{x}}\) is an eigenvector of \(A\) corresponding to an eigenvalue \(\lambda \).

a. Show that \(x\) is an eigenvector of \(5I - A\). What is the corresponding eigenvalue?

b. Show that \(x\) is an eigenvector of \(5I - 3A + {A^2}\). What is the corresponding eigenvalue?

Question: Let \(A = \left( {\begin{array}{*{20}{c}}{.5}&{.2}&{.3}\\{.3}&{.8}&{.3}\\{.2}&0&{.4}\end{array}} \right)\), \({{\rm{v}}_1} = \left( {\begin{array}{*{20}{c}}{.3}\\{.6}\\{.1}\end{array}} \right)\), \({{\rm{v}}_2} = \left( {\begin{array}{*{20}{c}}1\\{ - 3}\\2\end{array}} \right)\), \({{\rm{v}}_3} = \left( {\begin{array}{*{20}{c}}{ - 1}\\0\\1\end{array}} \right)\) and \({\rm{w}} = \left( {\begin{array}{*{20}{c}}1\\1\\1\end{array}} \right)\).

  1. Show that \({{\rm{v}}_1}\), \({{\rm{v}}_2}\), and \({{\rm{v}}_3}\) are eigenvectors of \(A\). (Note: \(A\) is the stochastic matrix studied in Example 3 of Section 4.9.)
  2. Let \({{\rm{x}}_0}\) be any vector in \({\mathbb{R}^3}\) with non-negative entries whose sum is 1. (In section 4.9, \({{\rm{x}}_0}\) was called a probability vector.) Explain why there are constants \({c_1}\), \({c_2}\), and \({c_3}\) such that \({{\rm{x}}_0} = {c_1}{{\rm{v}}_1} + {c_2}{{\rm{v}}_2} + {c_3}{{\rm{v}}_3}\). Compute \({{\rm{w}}^T}{{\rm{x}}_0}\), and deduce that \({c_1} = 1\).
  3. For \(k = 1,2, \ldots ,\) define \({{\rm{x}}_k} = {A^k}{{\rm{x}}_0}\), with \({{\rm{x}}_0}\) as in part (b). Show that \({{\rm{x}}_k} \to {{\rm{v}}_1}\) as \(k\) increases.

Let \({\bf{u}}\) be an eigenvector of \(A\) corresponding to an eigenvalue \(\lambda \), and let \(H\) be the line in \({\mathbb{R}^{\bf{n}}}\) through \({\bf{u}}\) and the origin.

  1. Explain why \(H\) is invariant under \(A\) in the sense that \(A{\bf{x}}\) is in \(H\) whenever \({\bf{x}}\) is in \(H\).
  2. Let \(K\) be a one-dimensional subspace of \({\mathbb{R}^{\bf{n}}}\) that is invariant under \(A\). Explain why \(K\) contains an eigenvector of \(A\).

Question: Exercises 9-14 require techniques section 3.1. Find the characteristic polynomial of each matrix, using either a cofactor expansion or the special formula for \(3 \times 3\) determinants described prior to Exercise 15-18 in Section 3.1. [Note: Finding the characteristic polynomial of a \(3 \times 3\) matrix is not easy to do with just row operations, because the variable \(\lambda \) is involved.

12. \(\left[ {\begin{array}{*{20}{c}}- 1&0&1\\- 3&4&1\\0&0&2\end{array}} \right]\)

(M) The MATLAB command roots\(\left( p \right)\) computes the roots of the polynomial equation \(p\left( t \right) = {\bf{0}}\). Read a MATLAB manual, and then describe the basic idea behind the algorithm for the roots command.
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