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Chapter 5: Q5.5-26E (page 267)

Let \(A\) be a real \(2 \times 2\) matrix with a complex eigenvalue \(\lambda = a - bi\)\(\left( {b \ne 0} \right)\) and an associated eigenvector \({\bf{v}}\) in \({\mathbb{}^2}\).

  1. Show that \(A({\mathop{\rm Re}\nolimits} {\bf{v}}) = a{\mathop{\rm Re}\nolimits} {\bf{v}} + b{\mathop{\rm Im}\nolimits} {\bf{v}}\) and \(A({\mathop{\rm Im}\nolimits} {\bf{v}}) = - b{\mathop{\rm Re}\nolimits} {\bf{v}} + a{\mathop{\rm Im}\nolimits} {\bf{v}}\). (Hint: Write \({\bf{v}} = {\mathop{\rm Re}\nolimits} {\bf{v}} + i{\mathop{\rm Im}\nolimits} {\bf{v}}\), and compute \(A{\bf{v}}\).)
  2. Verify that if \(P\) and \(C\) are given as in Theorem 9, then \(AP = PC\)

Short Answer

Expert verified
  1. \(A\left( {{\mathop{\rm Re}\nolimits} \left( {\bf{v}} \right)} \right) = {\mathop{\rm Re}\nolimits} \left( {A{\bf{v}}} \right) = a{\mathop{\rm Re}\nolimits} \left( {\bf{v}} \right) + b{\mathop{\rm Im}\nolimits} \left( {\bf{v}} \right)\)\(A\left( {{\mathop{\rm Im}\nolimits} \left( {\bf{v}} \right)} \right) = {\mathop{\rm Im}\nolimits} \left( {A{\bf{v}}} \right) = - b{\mathop{\rm Re}\nolimits} \left( {\bf{v}} \right) + aIm\left( {\bf{v}} \right)\)
  2. \(AP = \left( {A\left( {{\mathop{\rm Re}\nolimits} \left( {\bf{v}} \right)} \right)\,\,\,\,\,A\left( {{\mathop{\rm Im}\nolimits} \left( {\bf{v}} \right)} \right)} \right) = \left( {P\left( {\begin{aligned}{}a\\b\end{aligned}} \right)\;\;\;P\left( {\begin{aligned}{}{ - b}\\a\end{aligned}} \right)} \right) = P\left( {\begin{aligned}{}a&{}&{ - b}\\b&{}&a\end{aligned}} \right) = PC\)

Step by step solution

01

Formula of eigenvector 

Remember that an eigenvalue\(\lambda \)and an eigenvector\(x\)for a square matrix A satisfy the equation\(Ax = \lambda x\).

02

Proof

(a)

Given an eigenvector \(v\) and eigenvalue \(\lambda = a - bi\). Consider \(Ax = \lambda x\) and solve it as follows:

\(\begin{aligned}{}Av &= \lambda v\\ &= \left( {a - bi} \right)\left( {{\mathop{\rm Re}\nolimits} \left( v \right) + i{\mathop{\rm Im}\nolimits} \left( v \right)} \right)\\ &= \left( {a{\mathop{\rm Re}\nolimits} \left( v \right) + b{\mathop{\rm Im}\nolimits} \left( v \right)} \right) + i\left( {a{\mathop{\rm Im}\nolimits} \left( v \right) - b{\mathop{\rm Re}\nolimits} \left( v \right)} \right)\end{aligned}\)

So, from the above equation, it can be concluded that

\(\begin{aligned}{}A\left( {{\mathop{\rm Re}\nolimits} \left( v \right)} \right) = {\mathop{\rm Re}\nolimits} \left( {Av} \right) = a{\mathop{\rm Re}\nolimits} \left( v \right) + b{\mathop{\rm Im}\nolimits} \left( v \right)\\A\left( {{\mathop{\rm Re}\nolimits} \left( v \right)} \right) = {\mathop{\rm Im}\nolimits} \left( {Av} \right) = - b{\mathop{\rm Re}\nolimits} \left( v \right) + a{\mathop{\rm Im}\nolimits} \left( v \right)\end{aligned}\)

03

Verify that if P and C are given as in Theorem 9, then AP = PC. 

(b)

By theorem 9, let \(P = \left( {{\mathop{\rm Re}\nolimits} \left( v \right)\,\,\,\,\,\,\,\,{\mathop{\rm Im}\nolimits} \left( v \right)} \right)\)

From part (a),

\(\begin{aligned}{}A\left( {{\mathop{\rm Re}\nolimits} \left( v \right)} \right) &= P\left( {\begin{aligned}{}a\\b\end{aligned}} \right)\\A\left( {{\mathop{\rm Im}\nolimits} \left( v \right)} \right) &= P\left( {\begin{aligned}{}{ - b}\\a\end{aligned}} \right)\end{aligned}\)

Now, Solve AP as follows:

\(\begin{aligned}{}AP &= \left( {A\left( {{\mathop{\rm Re}\nolimits} \left( v \right)} \right)\,\,\,\,\,A\left( {{\mathop{\rm Im}\nolimits} \left( v \right)} \right)} \right)\\ &= \left( {P\left( {\begin{aligned}{}a\\b\end{aligned}} \right)\;\;\;P\left( {\begin{aligned}{}{ - b}\\a\end{aligned}} \right)} \right)\\ &= P\left( {\begin{aligned}{}a&{}&{ - b}\\b&{}&a\end{aligned}} \right)\\ &= PC\end{aligned}\)

Hence, it proved that \(AP = PC\).

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Most popular questions from this chapter

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 \(I\).)

Let\(T:{{\rm P}_2} \to {{\rm P}_3}\) be a linear transformation that maps a polynomial \({\bf{p}}\left( t \right)\) into the polynomial \(\left( {t + 5} \right){\bf{p}}\left( t \right)\).

  1. Find the image of\({\bf{p}}\left( t \right) = 2 - t + {t^2}\).
  2. Show that \(T\) is a linear transformation.
  3. Find the matrix for \(T\) relative to the bases \(\left\{ {1,t,{t^2}} \right\}\) and \(\left\{ {1,t,{t^2},{t^3}} \right\}\).

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 \(p\): \({C_p} = \left( {\begin{aligned}{*{20}{c}}{\bf{0}}&{\bf{1}}&{\bf{0}}&{...}&{\bf{0}}\\{\bf{0}}&{\bf{0}}&{\bf{1}}&{}&{\bf{0}}\\:&{}&{}&{}&:\\{\bf{0}}&{\bf{0}}&{\bf{0}}&{}&{\bf{1}}\\{ - {a_{\bf{0}}}}&{ - {a_{\bf{1}}}}&{ - {a_{\bf{2}}}}&{...}&{ - {a_{n - {\bf{1}}}}}\end{aligned}} \right)\).

19. Write the companion matrix \({C_p}\) for \(p\left( t \right) = {\bf{6}} - {\bf{5}}t + {t^{\bf{2}}}\), and then find the characteristic polynomial of \({C_p}\).

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\(B\).

Question: Let \(A = \left( {\begin{array}{*{20}{c}}{.6}&{.3}\\{.4}&{.7}\end{array}} \right)\), \({v_1} = \left( {\begin{array}{*{20}{c}}{3/7}\\{4/7}\end{array}} \right)\), \({x_0} = \left( {\begin{array}{*{20}{c}}{.5}\\{.5}\end{array}} \right)\). (Note: \(A\) is the stochastic matrix studied in Example 5 of Section 4.9.)

  1. Find a basic for \({\mathbb{R}^2}\) consisting of \({{\rm{v}}_1}\) and anther eigenvector \({{\rm{v}}_2}\) of \(A\).
  2. Verify that \({{\rm{x}}_0}\) may be written in the form \({{\rm{x}}_0} = {{\rm{v}}_1} + c{{\rm{v}}_2}\).
  3. For \(k = 1,2, \ldots \), define \({x_k} = {A^k}{x_0}\). Compute \({x_1}\) and \({x_2}\), and write a formula for \({x_k}\). Then show that \({{\bf{x}}_k} \to {{\bf{v}}_1}\) as \(k\) increases.
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