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Chapter 5: Q5.3-29E (page 267)

Question: A factorization \(A = PD{P^{ - {\bf{1}}}}\) is not unique. Demonstrate this for the matrix A in Example 2. With \({D_{\bf{1}}} = \left[ {\begin{array}{*{20}{c}}{\bf{3}}&{\bf{0}}\\{\bf{0}}&{\bf{5}}\end{array}} \right]\), use the information in Example 2 to find the matrix \({P_{\bf{1}}}\) such that \(A = {P_{\bf{1}}}{D_{\bf{1}}}P_{\bf{1}}^{ - {\bf{1}}}\).

Short Answer

Expert verified

The matrix \({P_1}\) is \(\left[ {\begin{array}{*{20}{c}}1&1\\{ - 2}&{ - 1}\end{array}} \right]\).

Step by step solution

01

Write the information given

The matrices from example 2 are,

\[A = \left[ {\begin{array}{*{20}{c}}7&2\\{ - 4}&1\end{array}} \right], P = \left[ {\begin{array}{*{20}{c}}1&1\\{ - 1}&{ - 2}\end{array}} \right], D = \left[ {\begin{array}{*{20}{c}}5&0\\0&3\end{array}} \right]\]

02

Find the matrix \({P_{\bf{1}}}\)

For eigenvalue 5, the eigenvector is \(\left[ {\begin{array}{*{20}{c}}1\\{ - 1}\end{array}} \right]\) and for eigenvalue 3, the eigenvector is \(\left[ {\begin{array}{*{20}{c}}1\\{ - 2}\end{array}} \right]\).

The matrix \({P_1}\) is such that \(A = {P_1}{D_1}P_1^{ - 1}\), so the matrix \({D_1}\) is:

\[{D_1} = \left[ {\begin{array}{*{20}{c}}3&0\\0&5\end{array}} \right]\]

Interchange the columns of P to find the matrix \({P_1}\).

\[{P_1} = \left[ {\begin{array}{*{20}{c}}1&1\\{ - 2}&{ - 1}\end{array}} \right]\]

03

Find the product \({P_{\bf{1}}}{D_{\bf{1}}}P_{\bf{1}}^{ - {\bf{1}}}\)

The product \({P_1}{D_1}P_1^{ - 1}\) can be calculated as follows:

\[\begin{array}{c}{P_1}{D_1}P_1^{ - 1} = \left[ {\begin{array}{*{20}{c}}1&1\\{ - 2}&{ - 1}\end{array}} \right]\left[ {\begin{array}{*{20}{c}}3&0\\0&5\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{ - 1}&{ - 1}\\2&1\end{array}} \right]\\ = \left[ {\begin{array}{*{20}{c}}7&2\\{ - 4}&1\end{array}} \right]\\ = A\end{array}\]

So, the matrix \({P_1}\) is \(\left[ {\begin{array}{*{20}{c}}1&1\\{ - 2}&{ - 1}\end{array}} \right]\).

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Most popular questions from this chapter

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\(B\).

Let\(D = \left\{ {{{\bf{d}}_1},{{\bf{d}}_2}} \right\}\) and \(B = \left\{ {{{\bf{b}}_1},{{\bf{b}}_2}} \right\}\) be bases for vector space \(V\) and \(W\), respectively. Let \(T:V \to W\) be a linear transformation with the property that

\(T\left( {{{\bf{d}}_1}} \right) = 2{{\bf{b}}_1} - 3{{\bf{b}}_2}\), \(T\left( {{{\bf{d}}_2}} \right) = - 4{{\bf{b}}_1} + 5{{\bf{b}}_2}\)

Find the matrix for \(T\) relative to \(D\), and\(B\).

[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.

20. \[A{\bf{ = }}\left[ {\begin{array}{*{20}{c}}{\bf{1}}&{\bf{2}}&{\bf{3}}&{\bf{2}}\\{\bf{2}}&{{\bf{12}}}&{{\bf{13}}}&{{\bf{11}}}\\{{\bf{ - 2}}}&{\bf{3}}&{\bf{0}}&{\bf{2}}\\{\bf{4}}&{\bf{5}}&{\bf{7}}&{\bf{2}}\end{array}} \right]\]

Apply the results of Exercise \({\bf{15}}\) to find the eigenvalues of the matrices \(\left( {\begin{aligned}{*{20}{c}}{\bf{1}}&{\bf{2}}&{\bf{2}}\\{\bf{2}}&{\bf{1}}&{\bf{2}}\\{\bf{2}}&{\bf{2}}&{\bf{1}}\end{aligned}} \right)\) and \(\left( {\begin{aligned}{*{20}{c}}{\bf{7}}&{\bf{3}}&{\bf{3}}&{\bf{3}}&{\bf{3}}\\{\bf{3}}&{\bf{7}}&{\bf{3}}&{\bf{3}}&{\bf{3}}\\{\bf{3}}&{\bf{3}}&{\bf{7}}&{\bf{3}}&{\bf{3}}\\{\bf{3}}&{\bf{3}}&{\bf{3}}&{\bf{7}}&{\bf{3}}\\{\bf{3}}&{\bf{3}}&{\bf{3}}&{\bf{3}}&{\bf{7}}\end{aligned}} \right)\).

Show that if \(A\) is diagonalizable, with all eigenvalues less than 1 in magnitude, then \({A^k}\) tends to the zero matrix as \(k \to \infty \). (Hint: Consider \({A^k}x\) where \(x\) represents any one of the columns of \(I\).)
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