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Chapter 5: Q32E (page 267)

Question: In Exercises 31 and 32, let A be the matrix of the linear transformation T. Without writing A, find an eigenvalue of A and describe the Eigenspace.

32. T is the transformation on \({\mathbb{R}^3}\) that rotates points about some line through the origin.

Short Answer

Expert verified

The Eigenvalue is \(1\) , and the Eigenspace is the axis of rotation.

Step by step solution

01

Given information

Here, \(T\) is the transformation on \({\mathbb{R}^3}\) and \(T\) rotates some points about some line through the origin.

02

Find Eigenvalue and Eigenspace of A

Consider \(l\) be the axis of rotation, now observe that any vector on this line will not change by this rotation, that is, \(T{\bf{v}} = {\bf{v}}\).

This implies that \(l\) is an eigenspace corresponding to the Eigenvalue \(\lambda = 1\).

Now, since the rotation does not change the vector's length, there may be one additional real Eigenvalue \(\lambda = - 1\).

Note that if this Eigenvalue exists, then this implies that after the rotation, the vector \({\bf{v}}\)becomes \( - {\bf{v}}\). This is possible only for rotation by \(180^\circ \)of any vector which is perpendicular to the axis \(l\).

Hence,If the rotation is by \(\alpha = 180^\circ \), then there is only one real Eigenvalue \(\lambda = 1\) , and the Eigenspace is the axis of rotation.

If, the rotation is by \(\alpha \ne 180^\circ \), then there is an additional real Eigenvalue \(\lambda = - 1\) , and its Eigenspace is the plane that passes through the origin and also perpendicular to the axis of rotation.

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

Consider an invertible n 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)=-Ax鈫�(t)

Question 19: Let \(A\) be an \(n \times n\) matrix, and suppose A has \(n\) real eigenvalues, \({\lambda _1},...,{\lambda _n}\), repeated according to multiplicities, so that \(\det \left( {A - \lambda I} \right) = \left( {{\lambda _1} - \lambda } \right)\left( {{\lambda _2} - \lambda } \right) \ldots \left( {{\lambda _n} - \lambda } \right)\) . Explain why \(\det A\) is the product of the n eigenvalues of A. (This result is true for any square matrix when complex eigenvalues are considered.)

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

8. \(\left[ {\begin{array}{*{20}{c}}7&- 2\\2&3\end{array}} \right]\)

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

8. \(\left( {\begin{array}{*{20}{c}}{\bf{5}}&{\bf{1}}\\{\bf{0}}&{\bf{5}}\end{array}} \right)\)
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