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Q5.2-30E

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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

Question: Let \(A = \left( {\begin{array}{*{20}{c}}{ - 6}&{28}&{21}\\4&{ - 15}&{ - 12}\\{ - 8}&a&{25}\end{array}} \right)\). For each value of \(a\) in the set \(\left\{ {32,31.9,31.8,32.1,32.2} \right\}\), compute the characteristic polynomial of \(A\) and the eigenvalues. In each case, create a graph of the characteristic polynomial \(p\left( t \right) = \det \left( {A - tI} \right)\) for \(0 \le t \le 3\). If possible, construct all graphs on one coordinate system. Describe how the graphs reveal the changes in the eigenvalues of \(a\) changes.

Characteristic polynomial and eigenvalues are:

\[a\]

Characteristic polynomial

Eigenvalues

\[31.8\]

\[ - .4 - 2.6t + 4{t^2} - {t^3}\]

\[3.1279,1, - .1279\]

\[31.9\]

\[.8 - 3.8t + 4{t^2} - {t^3}\]

\[2.7042,1,.2958\]

\[32.0\]

\[2 - 5t + 4{t^2} - {t^3}\]

\[2,1,1\]

\[32.1\]

\[3.2 - 6.2t + 4{t^2} - {t^3}\]

\[1.5 \pm .9747i,1\]

\[32.2\]

\[4.4 - 7.4t + 4{t^2} - {t^3}\]

\[1.5 \pm 1.4663i,1\]

The graph of the characteristic polynomial is shown below:

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1: Determine characteristic polynomial and eigenvalues for the matrix A

Consider the matrix \(A = \left( {\begin{array}{*{20}{c}}{ - 6}&{28}&{21}\\4&{ - 15}&{ - 12}\\{ - 8}&a&{25}\end{array}} \right)\). Consider \(a = 32\).

Use the following command in the MATLAB to find the characteristic polynomial and eigenvalues of the matrix.

\[\begin{array}{l} > > {\rm{A}} = \left( {\begin{array}{*{20}{c}}{ - 6}&{28}&{21;}&4&{ - 15}&{ - 12;}\\{ - 8}&{32}&{25;}&{}&{}&{}\end{array}} \right);\\ > > {\rm{p}} = {\rm{poly}}\left( {\rm{A}} \right);\\ > > {\rm{eig}} = {\rm{eign}}\left( {\rm{A}} \right)\end{array}\]

So, the characteristic polynomial and eigenvalues of A is \(p = 2 - 5t + 4{t^2} - {t^3}\), \({\rm{eig}} = \left\{ {2,1,1} \right\}\).

Step 2: Determine characteristic polynomial and eigenvalues for the matrix A for each value of set a

Use the following command in MATLAB to find the characteristic polynomial and eigenvalues of the matrix.

\[\begin{array}{l} > > {\rm{A}} = \left( {\begin{array}{*{20}{c}}{ - 6}&{28}&{21;}&4&{ - 15}&{ - 12;}\\{ - 8}&a&{25;}&{}&{}&{}\end{array}} \right);\\ > > {\rm{p}} = {\rm{poly}}\left( {\rm{A}} \right);\\ > > {\rm{eig}} = {\rm{eign}}\left( {\rm{A}} \right)\end{array}\]

So, the characteristic polynomial and eigenvalues of A is shown below:

\[a\]

Characteristic polynomial

Eigenvalues

\[31.8\]

\[ - .4 - 2.6t + 4{t^2} - {t^3}\]

\[3.1279,1, - .1279\]

\[31.9\]

\[.8 - 3.8t + 4{t^2} - {t^3}\]

\[2.7042,1,.2958\]

\[32.0\]

\[2 - 5t + 4{t^2} - {t^3}\]

\[2,1,1\]

\[32.1\]

\[3.2 - 6.2t + 4{t^2} - {t^3}\]

\[1.5 \pm .9747i,1\]

\[32.2\]

\[4.4 - 7.4t + 4{t^2} - {t^3}\]

\[1.5 \pm 1.4663i,1\]

Step 3: Plot the graph of characteristic polynomial

The procedure to draw the graph of the above equation by using the graphing calculator is as follows:

Draw the graph of the function \(f\left( t \right) = - .4 - 2.6t + 4{t^2} - {t^3}\), \(g\left( t \right) = .8 - 3.8t + 4{t^2} - {t^3}\), \(k\left( t \right) = 2 - 5t + 4{t^2} - {t^3}\), \(x\left( t \right) = 3.2 - 6.2t + 4{t^2} - {t^3}\) and \(y\left( t \right) = 4.4 - 7.4t + 4{t^2} - {t^3}\) by using the graphing calculator as shown below:

  1. Open the graphing calculator. Select the “STAT PLOT” and enter the equation \( - .4 - 2.6t + 4{t^2} - {t^3}\) in the \({Y_1}\) tab.
  2. Select the “STAT PLOT” and enter the equation \(.8 - 3.8t + 4{t^2} - {t^3}\) in the \({Y_2}\) tab.
  3. Select the “STAT PLOT” and enter the equation \(2 - 5t + 4{t^2} - {t^3}\) in the \({Y_3}\) tab.
  4. Select the “STAT PLOT” and enter the equation \(3.2 - 6.2t + 4{t^2} - {t^3}\) in the \({Y_4}\) tab.
  5. Select the “STAT PLOT” and enter the equation \(4.4 - 7.4t + 4{t^2} - {t^3}\) in the \({Y_5}\) tab.
  6. Enter the “GRAPH” button in the graphing calculator.

Visualizations of graphs of the functions stated above are shown below:

Most popular questions for Math Textbooks

Question 18: It can be shown that the algebraic multiplicity of an eigenvalue \(\lambda \) is always greater than or equal to the dimension of the eigenspace corresponding to \(\lambda \). Find \(h\) in the matrix \(A\) below such that the eigenspace for \(\lambda = 5\) is two-dimensional:

\[A = \left[ {\begin{array}{*{20}{c}}5&{ - 2}&6&{ - 1}\\0&3&h&0\\0&0&5&4\\0&0&0&1\end{array}} \right]\]

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 \(A\).

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 \({\mathbb{R}^4}\).

Compute the quantities in Exercises 1-8 using the vectors

\({\mathop{\rm u}\nolimits} = \left( {\begin{array}{*{20}{c}}{ - 1}\\2\end{array}} \right),{\rm{ }}{\mathop{\rm v}\nolimits} = \left( {\begin{array}{*{20}{c}}4\\6\end{array}} \right),{\rm{ }}{\mathop{\rm w}\nolimits} = \left( {\begin{array}{*{20}{c}}3\\{ - 1}\\{ - 5}\end{array}} \right),{\rm{ }}{\mathop{\rm x}\nolimits} = \left( {\begin{array}{*{20}{c}}6\\{ - 2}\\3\end{array}} \right)\)

3. \(\frac{1}{{{\mathop{\rm w}\nolimits} \cdot {\mathop{\rm w}\nolimits} }}{\mathop{\rm w}\nolimits} \)

Define \(T:{{\rm P}_2} \to {\mathbb{R}^3}\) by \(T\left( {\bf{p}} \right) = \left( {\begin{aligned}{{\bf{p}}\left( { - 1} \right)}\\{{\bf{p}}\left( 0 \right)}\\{{\bf{p}}\left( 1 \right)}\end{aligned}} \right)\).

  1. Find the image under\(T\)of\({\bf{p}}\left( t \right) = 5 + 3t\).
  2. Show that \(T\) is a linear transformation.
  3. Find the matrix for \(T\) relative to the basis \(\left\{ {1,t,{t^2}} \right\}\)for \({{\rm P}_2}\)and the standard basis for \({\mathbb{R}^3}\).
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