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Chapter 5: Q5.5-28E (page 267)

In Exercises 27 and 28, 铿乶d a factorization of the given matrix A in the form \(A = PC{P^{ - 1}}\),where \(C\) is a block-diagonal matrix with \(2 \times 2\) blocks of the form shown in Example 6. (For each conjugate pair of eigenvalues, use the real and imaginary parts of one eigenvector in \({\mathbb{}^4}\) to create two columns of P.)

28. \(\left( {\begin{aligned}{}{ - 1.4}&{}&{ - 2.0}&{}&{ - 2.0}&{}&{ - 2.0}\\{ - 1.3}&{}&{ - .8}&{}&{ - .1}&{}&{ - .6}\\{.3}&{}&{ - 1.9}&{}&{ - 1.6}&{}&{ - 1.4}\\{2.0}&{}&{3.3}&{}&{2.3}&{}&{2.6}\end{aligned}} \right)\)

Short Answer

Expert verified

The factorization is \(P = \left( {\begin{aligned}{}{ - 1}&{}&{ - 1}&{}&0&{}&0\\{ - 1}&{}&1&{}&{ - 1}&{}&{ - 1}\\1&{}&{ - 1}&{}&{ - 1}&{}&1\\1&{}&0&{}&2&{}&0\end{aligned}} \right)\), \(C = \left( {\begin{aligned}{}{ - .4}&{}&{ - 1}&{}&0&{}&0\\1&{}&{ - .4}&{}&0&{}&0\\0&{}&0&{}&{ - .2}&{}&{ - .5}\\0&{}&0&{}&{.5}&{}&{ - .2}\end{aligned}} \right)\).

Step by step solution

01

Input the matrix

\({\rm{ > > A = }}\left( {{\rm{0}}{\rm{.7 1}}{\rm{.1 2 1}}{\rm{.7; - 2 - 4 - 8}}{\rm{.6 - 7}}{\rm{.4; 0 - 0}}{\rm{.5 - 1 - 1; 1 2}}{\rm{.8 6 5}}{\rm{.3}}} \right)\)

02

Get the expression of command for getting the Eigenvalue(s) of A

We use the following command to get the eigenvalue of the matrix \(A\).

\({\rm{ev}} = {\mathop{\rm eig}\nolimits} ({\bf{A}}) = ( - .4 + {\rm{i}}, - .4 - {\rm{i}}, - .2 + .5{\rm{i}}, - .2 - .5{\rm{i}})\)

For \(\lambda = - .4 - i\), we find the value of the operand in this case to find the eigenvalues.

\({\mathop{\rm nulbasis}\nolimits} ({\bf{A}} - {\mathop{\rm ev}\nolimits} ({\bf{2}})*{\mathop{\rm eye}\nolimits} (4)) = \left( {\begin{aligned}{}{ - 1 - i}\\{ - 1 + i}\\{1 - i}\\1\end{aligned}} \right)\)

\({{\bf{v}}_1} = \left( {\begin{aligned}{}{ - 1 - i}\\{ - 1 + i}\\{1 - i}\\1\end{aligned}} \right)\)

For \(\lambda = - .2 - .5i\), we find the value of the operand in this case to find the eigenvalues.

\({\mathop{\rm nulbasis}\nolimits} ({\bf{A}} - {\mathop{\rm ev}\nolimits} (4)*{\mathop{\rm eye}\nolimits} (4)) = \left( {\begin{aligned}{}0\\{ - 1 - i}\\{ - 1 + i}\\2\end{aligned}} \right)\)

\({{\bf{v}}_2} = \left( {\begin{aligned}{}0\\{ - 1 - i}\\{ - 1 + i}\\2\end{aligned}} \right)\)

Now, by theorem 9:

\(P = \left( {\begin{aligned}{}{{\mathop{\rm Re}\nolimits} {v_1}}&{{\mathop{\rm Im}\nolimits} {v_1}}&{{\mathop{\rm Re}\nolimits} {v_2}}&{{\mathop{\rm Im}\nolimits} {v_2}}\end{aligned}} \right) = \left( {\begin{aligned}{}{ - 1}&{}&{ - 1}&{}&0&{}&0\\{ - 1}&{}&1&{}&{ - 1}&{}&{ - 1}\\1&{}&{ - 1}&{}&{ - 1}&{}&1\\1&{}&0&{}&2&{}&0\end{aligned}} \right)\)and \(C = \left( {\begin{aligned}{}{ - .4}&{}&{ - 1}&{}&0&{}&0\\1&{}&{ - .4}&{}&0&{}&0\\0&{}&0&{}&{ - .2}&{}&{ - .5}\\0&{}&0&{}&{.5}&{}&{ - .2}\end{aligned}} \right)\).

Hence, \(P = \left( {\begin{aligned}{}{ - 1}&{}&{ - 1}&{}&0&{}&0\\{ - 1}&{}&1&{}&{ - 1}&{}&{ - 1}\\1&{}&{ - 1}&{}&{ - 1}&{}&1\\1&{}&0&{}&2&{}&0\end{aligned}} \right),C = \left( {\begin{aligned}{}{ - .4}&{}&{ - 1}&{}&0&{}&0\\1&{}&{ - .4}&{}&0&{}&0\\0&{}&0&{}&{ - .2}&{}&{ - .5}\\0&{}&0&{}&{.5}&{}&{ - .2}\end{aligned}} \right)\).

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Most popular questions from this chapter

Exercises 19鈥�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 \(p\).

Question: In Exercises \({\bf{5}}\) and \({\bf{6}}\), the matrix \(A\) is factored in the form \(PD{P^{ - {\bf{1}}}}\). Use the Diagonalization Theorem to find the eigenvalues of \(A\) and a basis for each eigenspace.

6. \(\left( {\begin{array}{*{20}{c}}{\bf{4}}&{\bf{0}}&{{\bf{ - 2}}}\\{\bf{2}}&{\bf{5}}&{\bf{4}}\\{\bf{0}}&{\bf{0}}&{\bf{5}}\end{array}} \right){\bf{ = }}\left( {\begin{array}{*{20}{c}}{{\bf{ - 2}}}&{\bf{0}}&{{\bf{ - 1}}}\\{\bf{0}}&{\bf{1}}&{\bf{2}}\\{\bf{1}}&{\bf{0}}&{\bf{0}}\end{array}} \right)\left( {\begin{array}{*{20}{c}}{\bf{5}}&{\bf{0}}&{\bf{0}}\\{\bf{0}}&{\bf{5}}&{\bf{0}}\\{\bf{0}}&{\bf{0}}&4\end{array}} \right)\left( {\begin{array}{*{20}{c}}{\bf{0}}&{\bf{0}}&{\bf{1}}\\{\bf{2}}&{\bf{1}}&{\bf{4}}\\{{\bf{ - 1}}}&{\bf{0}}&{{\bf{ - 2}}}\end{array}} \right)\)

Let \(A{\bf{ = }}\left( {\begin{aligned}{*{20}{c}}{{a_{{\bf{11}}}}}&{{a_{{\bf{12}}}}}\\{{a_{{\bf{21}}}}}&{{a_{{\bf{22}}}}}\end{aligned}} \right)\). Recall from Exercise \({\bf{25}}\) in Section \({\bf{5}}{\bf{.4}}\) that \({\rm{tr}}\;A\) (the trace of \(A\)) is the sum of the diagonal entries in \(A\). Show that the characteristic polynomial of \(A\) is \({\lambda ^2} - \left( {{\rm{tr}}A} \right)\lambda + \det A\). Then show that the eigenvalues of a \({\bf{2 \times 2}}\) matrix \(A\) are both real if and only if \(\det A \le {\left( {\frac{{{\rm{tr}}A}}{2}} \right)^2}\).

Let\(\varepsilon = \left\{ {{{\bf{e}}_1},{{\bf{e}}_2},{{\bf{e}}_3}} \right\}\) be the standard basis for \({\mathbb{R}^3}\),\(B = \left\{ {{{\bf{b}}_1},{{\bf{b}}_2},{{\bf{b}}_3}} \right\}\) be a basis for a vector space \(V\) and\(T:{\mathbb{R}^3} \to V\) be a linear transformation with the property that

\(T\left( {{x_1},{x_2},{x_3}} \right) = \left( {{x_3} - {x_2}} \right){{\bf{b}}_1} - \left( {{x_1} - {x_3}} \right){{\bf{b}}_2} + \left( {{x_1} - {x_2}} \right){{\bf{b}}_3}\)

  1. Compute\(T\left( {{{\bf{e}}_1}} \right)\), \(T\left( {{{\bf{e}}_2}} \right)\) and \(T\left( {{{\bf{e}}_3}} \right)\).
  2. Compute \({\left( {T\left( {{{\bf{e}}_1}} \right)} \right)_B}\), \({\left( {T\left( {{{\bf{e}}_2}} \right)} \right)_B}\) and \({\left( {T\left( {{{\bf{e}}_3}} \right)} \right)_B}\).
  3. Find the matrix for \(T\) relative to \(\varepsilon \), and\(B\).

Show that if \({\bf{x}}\) is an eigenvector of the matrix product \(AB\) and \(B{\rm{x}} \ne 0\), then \(B{\rm{x}}\) is an eigenvector of\(BA\).
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