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Chapter 7: Q4E (page 380)

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

4.A=[鈭�0.9鈭�0.40.4鈭�0.9],

Short Answer

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Stable

Step by step solution

01

Determine whether the zero state is a stable equilibrant

We solve

det(A鈭�位I)=0|鈭�0.9鈭�位鈭�0.40.4鈭�0.9鈭�位|=0(鈭�0.9鈭�位)2+0.16=0位2+1.8位+0.97=0位1,2=鈭�1.8卤3.24鈭�3.882=鈭�0.9卤0.4i

Since|位1|=|位2|=0.97<1 then the equilibrium is stable.

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Most popular questions from this chapter

27: a. Based on your answers in Exercises 24 and 25, find closed formulas for the components of the dynamical system

x炉(t+1)=[0.50.250.50.75]x炉(t)

with initial value x0鈫�=e1鈫�. Then do the same for the initial value x0鈫�=e2鈫�. Sketch the two trajectories.

b. Consider the matrix

A=[0.50.250.50.75]

.

Using technology, compute some powers of the matrix A, say, A2, A5, A10, . . . .What do you observe? Diagonalize matrix Ato prove your conjecture. (Do not use Theorem 2.3.11, which we have not proven

yet.)

c. If A=[abcd]

is an arbitrary positive transition matrix, what can you say about the powers Atas t goes to infinity? Your result proves Theorem 2.3.11c for the special case of a positive transition matrix of size 2 脳 2.

consider an eigenvalue 位0of ann脳nmatrix A. we are told that the algebraic multiplicity of exceeds 1.Show thatf'(位0)=0(i.e.., the derivative of the characteristic polynomial of A vanishes are位0).

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

Find a basis of the linear space Vof all2脳2matrices Afor which[1-3] is an eigenvector, and thus determine the dimension of V.

If v鈬赌is any nonzero vector in R2 , what is the dimension of the space Vof all 2脳2matrices for which v鈬赌is an eigenvector?
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