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Chapter 7: Q2E (page 380)

For the matrices A in Exercises 1 through 10 , determine whether the zero state is a stable equilibriunt of the dynamical systemx鈫�(i+1)=Ax鈫�(i)

2.A=[鈭�1.1000.9],

Short Answer

Expert verified

Unstable

Step by step solution

01

Determine whether the zero state is a stable equilibriunt

The matrix A is diagonal, so we have

At=[鈭�1.1t000.9],Atx0=[1.1t000.9t][x1x2]=[1.1tx10.9tx2],

Which leads to the conclusion that limt鈫�鈭�Atx0does not exist.

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Most popular questions from this chapter

Arguing geometrically, find all eigenvectors and eigenvalues of the linear transformations in Exercises 15 through 22. In each case, find an eigenbasis if you can, and thus determine whether the given transformation is diagonalizable.

Reflection about a line L inR2.

Is v鈬赌an eigenvector ofA+2In? If so, what is the eigenvalue?

Give an example of a matrixAof rank 1 that fails to be diagonalizable.

Arguing geometrically, find all eigenvectors and eigenvalues of the linear transformations in Exercises 15 through 22. In each case, find an eigenbasis if you can, and thus determine whether the given transformation is diagonalizable.

Orthogonal projection onto a line L in R3.

If a vector v鈬赌is an eigenvector of both Aand B, isv鈬赌necessarily an eigenvector of A+B?
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