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Chapter 7: Q3E (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)

3.A=[0.80.7鈭�0.70.8],

Short Answer

Expert verified

Unstable

Step by step solution

01

Determine whether the zero state is a stable equilibrant

We solve

det(A鈭�位I)=0|0.8鈭�位0.7鈭�0.70.8鈭�位|=0(0.8鈭�位)2+0.49=0位2鈭�1.6位+1.13=0位1,2=1.6卤2.56鈭�4.522=0.8卤0.7i

Since|位1|=|位2|=1.13鈮�1, then, the equilibrium is unstable.

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Most popular questions from this chapter

Consider the matrix A=|abbc| where a, b, and c are nonzero constants. For which values of a, b, and c does A have two distinct eigenvalues?

In all parts of this problem, let V be the linear space of all 2 脳 2 matrices for which [12]is an eigenvector.

(a) Find a basis of V and thus determine the dimension of V.

(b) Consider the linear transformation T (A) = A[12] from V to R2. Find a basis of the image of Tand a basis of the kernel of T. Determine the rank of T .

(c) Consider the linear transformation L(A) = A[13] from V to R2. Find a basis of the image of L and a basis of the kernel of L. Determine the rank of L.

For a given eigenvalue, find a basis of the associated eigensspace .use the geometric multiplicities of the eigenvalues to determine whether a matrix is diagonalizable.

For each of the matrices A in Exercise1 through 20,find all (real) eigenvalues. Then find a basis of each eigenspaces ,and diagonalize A, if you can. Do not use technology.

[6327]

A. Find the characteristic polynomial of the matrix

A=[010001abc]

B. Can you find a 3脳3matrix whose characteristic polynomial is-位2+17位2-5位+蟿蟿

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).
See all solutions

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