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Chapter 5: Q26E (page 267)

Question: Show that if \({A^{\bf{2}}}\) is the zero matrix, then the only eigenvalue of A is 0.

Short Answer

Expert verified

If the matrix \({A^2}\) is zero, then each eigenvalue of A is zero.

Step by step solution

01

Write the given information

For matrix A, the matrix \({A^2}\) is 0.

02

Check for the eigenvalue of \({A^{ - {\bf{1}}}}\)

As \({A^2}\) is a zero matrix, so the equation \(A{\bf{x}} = \lambda {\bf{x}}\) and \(x \ne 0\) can be represented as;

\(\begin{array}{c}{A^2}{\bf{x}} = A\left( {A{\bf{x}}} \right)\\ = A\left( {\lambda {\bf{x}}} \right)\\ = \lambda A{\bf{x}}\\ = {\lambda ^2}{\bf{x}}\end{array}\)

As x is nonzero, so \(\lambda \) must be zero.

Therefore, the eigenvalue of A is zero.

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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 \(\left( {{\bf{ - 2,}}\;{\bf{1,}}\;{\bf{2}}} \right)\).

12. \(\left( {\begin{array}{*{20}{c}}{\bf{4}}&{\bf{2}}&{\bf{2}}\\{\bf{2}}&{\bf{4}}&{\bf{2}}\\{\bf{2}}&{\bf{2}}&{\bf{4}}\end{array}} \right)\)

Question: Construct a random integer-valued \(4 \times 4\) matrix \(A\), and verify that \(A\) and \({A^T}\) have the same characteristic polynomial (the same eigenvalues with the same multiplicities). Do \(A\) and \({A^T}\) have the same eigenvectors? Make the same analysis of a \(5 \times 5\) matrix. Report the matrices and your conclusions.

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

x鈫�(t+1)=(A-2In)x鈫�(t)

In Exercises 9鈥�16, find a basis for the eigenspace corresponding to each listed eigenvalue.

10. \(A = \left( {\begin{array}{*{20}{c}}{10}&{ - 9}\\4&{ - 2}\end{array}} \right)\), \(\lambda = 4\)

For the matrices A in Exercises 1 through 12, find closed formulas for , where t is an arbitrary positive integer. Follow the strategy outlined in Theorem 7.4.2 and illustrated in Example 2. In Exercises 9 though 12, feel free to use technology.

1.A=1203
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