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Chapter 5: Q5.3-28E (page 267)

Question: Show that if A has n linearly independent eigenvectors, then so does \({A^T}\). [Hint: Use the diagonalization theorem.]

Short Answer

Expert verified

It is proved that the matrix \({A^T}\) is diagonalizable and the columns of matrix Q are n linearly independent eigenvectors of \({A^T}\).

Step by step solution

01

Write the diagonalization theorem

If the matrix A has n linearly independent vectors, then by the diagonalization theorem, \(A = PD{P^{ - 1}}\).

The matrix P is invertible and matrix D is diagonalizable.

02

Check if transpose of A has n linearly independent vector

Using the properties of transpose,

\[\begin{array}{c}{A^T} = {\left( {PD{P^{ - 1}}} \right)^T}\\ = {\left( {{P^{ - 1}}} \right)^T}{D^T}{P^T}\\ = {\left( {{P^T}} \right)^{ - 1}}D{P^T}\\ = QD{Q^{ - 1}}\end{array}\]

Here, \(Q = {\left( {{P^T}} \right)^{ - 1}}\). Thus, the matrix \({A^T}\) is diagonalizable and the columns of matrix Q are n linearly independent eigenvectors of \({A^T}\).

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Most popular questions from this chapter

Question: Diagonalize the matrices in Exercises \({\bf{7--20}}\), if possible. The eigenvalues for Exercises \({\bf{11--16}}\) are as follows:\(\left( {{\bf{11}}} \right)\lambda {\bf{ = 1,2,3}}\); \(\left( {{\bf{12}}} \right)\lambda {\bf{ = 2,8}}\); \(\left( {{\bf{13}}} \right)\lambda {\bf{ = 5,1}}\); \(\left( {{\bf{14}}} \right)\lambda {\bf{ = 5,4}}\); \(\left( {{\bf{15}}} \right)\lambda {\bf{ = 3,1}}\); \(\left( {{\bf{16}}} \right)\lambda {\bf{ = 2,1}}\). For exercise \({\bf{18}}\), one eigenvalue is \(\lambda {\bf{ = 5}}\) and one eigenvector is \(\left( {{\bf{ - 2,}}\;{\bf{1,}}\;{\bf{2}}} \right)\).

15. \(\left( {\begin{array}{*{20}{c}}{\bf{7}}&{\bf{4}}&{{\bf{16}}}\\{\bf{2}}&{\bf{5}}&{\bf{8}}\\{{\bf{ - 2}}}&{{\bf{ - 2}}}&{{\bf{ - 5}}}\end{array}} \right)\)

Consider an invertiblen × n matrix A such that the zero state is a stable equilibrium of the dynamical system x→(t+1)=Ax→(t)What can you say about the stability of the systems

x→(t+1)=(A-2In)x→(t)

For the matrix A, find real closed formulas for the trajectory x→(t+1)=Ax¯(t) where x→=[01]. Draw a rough sketchA=[7-156-11]

Exercises 19–23 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 \(p\): \({C_p} = \left( {\begin{aligned}{*{20}{c}}{\bf{0}}&{\bf{1}}&{\bf{0}}&{...}&{\bf{0}}\\{\bf{0}}&{\bf{0}}&{\bf{1}}&{}&{\bf{0}}\\:&{}&{}&{}&:\\{\bf{0}}&{\bf{0}}&{\bf{0}}&{}&{\bf{1}}\\{ - {a_{\bf{0}}}}&{ - {a_{\bf{1}}}}&{ - {a_{\bf{2}}}}&{...}&{ - {a_{n - {\bf{1}}}}}\end{aligned}} \right)\).

20. Let \(p\left( t \right){\bf{ = }}\left( {t{\bf{ - 2}}} \right)\left( {t{\bf{ - 3}}} \right)\left( {t{\bf{ - 4}}} \right){\bf{ = - 24 + 26}}t{\bf{ - 9}}{t^{\bf{2}}}{\bf{ + }}{t^{\bf{3}}}\). Write the companion matrix for \(p\left( t \right)\), and use techniques from chapter \({\bf{3}}\) to find the characteristic polynomial.

Use Exercise 12 to find the eigenvalues of the matrices in Exercises 13 and 14.

13. \(A = \left( {\begin{array}{*{20}{c}}3&{ - 2}&8\\0&5&{ - 2}\\0&{ - 4}&3\end{array}} \right)\)
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