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Chapter 5: Q5.4-31E (page 267)

Find the \(B\) matrix for the transformation\({\rm{x}} \mapsto A{\rm{x}}\), when \(B = \left\{ {{b_1},{b_2},{b_3}} \right\}\).

\(A = \left( {\begin{aligned}{}{ - 7}&{ - 48}&{ - 16}\\1&{14}&6\\{ - 3}&{ - 45}&{ - 19}\end{aligned}} \right)\),

\({b_1} = \left( {\begin{aligned}{}{ - 3}\\1\\{ - 3}\end{aligned}} \right)\), \({b_2} = \left( {\begin{aligned}{}{ - 2}\\1\\{ - 3}\end{aligned}} \right)\), \({b_3} = \left( {\begin{aligned}{}{\,\,\,3}\\{ - 1}\\{\,\,\,0}\end{aligned}} \right)\)

Short Answer

Expert verified

The B matrix of the transformation \({\rm{x}} \mapsto A{\rm{x}}\)is \(D = {P^{ - 1}}AP\), which is \(\left( {\begin{aligned}{}{ - 7}&{}&{ - 2}&{}&{ - 6}\\0&{}&{ - 4}&{}&{ - 6}\\0&{}&0&{}&{ - 1}\end{aligned}} \right)\)

Step by step solution

01

Find P matrix

Let \(P\) is an invertible matrix and \(D\) is the diagonal matrix, such that the matrix A can be written as follows:

\(A = PD{P^{ - 1}}\)

Where the columns of the \(P\) matrix are the same as that of basis \(B\). So, the matrix \(P\) can be written as:

\(\begin{aligned}{c}P &= \left( {{{\rm{b}}_1}\,\,{{\rm{b}}_2}\,\,{{\rm{b}}_3}} \right)\\ &= \left( {\begin{aligned}{}{ - 3}&{}&{ - 2}&{}&3\\1&{}&1&{}&{ - 1}\\{ - 3}&{}&{ - 3}&{}&0\end{aligned}} \right)\end{aligned}\)

02

Find D matrix

As \(A = PD{P^{ - 1}}\) then the diagonal matrix \(D\) can be obtained as \(D = {P^{ - 1}}AP\). To find \(D = {P^{ - 1}}AP\), first, find \({P^{ - 1}}\)using the cofactors and determinant method.

\(\begin{aligned}{c}P &= \left( {\begin{aligned}{}{ - 3}&{}&{ - 2}&{}&3\\1&{}&1&{}&{ - 1}\\{ - 3}&{}&{ - 3}&{}&0\end{aligned}} \right)\\{P^{ - 1}} &= \left( {\begin{aligned}{}{ - 1}&{}&{ - 3}&{}&{ - 1/3}\\{\,\,1}&{}&{\,\,3}&{}&{\,0}\\{\,\,0}&{}&{ - 1}&{}&{ - 1/3}\end{aligned}} \right)\end{aligned}\)

Now find the matrix \(D\) as follows:

\(\begin{aligned}{c}D &= {P^{ - 1}}AP\\ &= \left( {\begin{aligned}{}{ - 1}&{}&{ - 3}&{}&{ - 1/3}\\{\,\,1}&{}&{\,\,3}&{}&{\,0}\\{\,\,0}&{}&{ - 1}&{}&{ - 1/3}\end{aligned}} \right)\left( {\begin{aligned}{}{ - 7}&{}&{ - 48}&{}&{ - 16}\\1&{}&{14}&{}&6\\{ - 3}&{}&{ - 45}&{}&{ - 19}\end{aligned}} \right)\left( {\begin{aligned}{}{ - 3}&{}&{ - 2}&{}&3\\1&{}&1&{}&{ - 1}\\{ - 3}&{}&{ - 3}&{}&0\end{aligned}} \right)\\ &= \left( {\begin{aligned}{}{ - 7}&{}&{ - 2}&{}&{ - 6}\\0&{}&{ - 4}&{}&{ - 6}\\0&{}&0&{}&{ - 1}\end{aligned}} \right)\end{aligned}\)

The B matrix of the transformation \({\rm{x}} \mapsto A{\rm{x}}\) is \(D = {P^{ - 1}}AP\), which is \(\left( {\begin{aligned}{}{ - 7}&{}&{ - 2}&{}&{ - 6}\\0&{}&{ - 4}&{}&{ - 6}\\0&{}&0&{}&{ - 1}\end{aligned}} \right)\).

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Most popular questions from this chapter

Question: In Exercises 21 and 22, \(A\) and \(B\) are \(n \times n\) matrices. Mark each statement True or False. Justify each answer.

  1. The determinant of \(A\) is the product of the diagonal entries in \(A\).
  2. An elementary row operation on \(A\) does not change the determinant.
  3. \(\left( {\det A} \right)\left( {\det B} \right) = \det AB\)
  4. If \(\lambda + 5\) is a factor of the characteristic polynomial of \(A\), then 5 is an eigenvalue of \(A\).

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

Let\(G = \left( {\begin{aligned}{*{20}{c}}A&X\\{\bf{0}}&B\end{aligned}} \right)\). Use formula\(\left( {\bf{1}} \right)\)for the determinant in section\({\bf{5}}{\bf{.2}}\)to explain why\(\det G = \left( {\det A} \right)\left( {\det B} \right)\). From this, deduce that the characteristic polynomial of\(G\)is the product of the characteristic polynomials of\(A\)and\(B\).

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: Find the characteristic polynomial and the eigenvalues of the matrices in Exercises 1-8.

  1. \(\left[ {\begin{array}{*{20}{c}}2&7\\7&2\end{array}} \right]\)
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