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Chapter 7: Q25E (page 372)

Find all complex eigenvalues of the matrices in Exercises 20 through 26 (including the real ones, of course). Do not use technology. Show all your work.

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Short Answer

Expert verified

The solution for all complex eigenvalues is位1,2=卤1,位3,4=卤1,位3,4=卤i

Step by step solution

01

 Step 1: Define eigenvalue

Eigenvalues are a set of specialized scales associated with a system of linear equations. The corresponding eigenvalue, often denoted by.

02

 Step 2: Find all complex eigenvalues

Consider the equation detA-位濒=0,

-位0011-位0001-位0001-位=0

-位.(-位)3-1.1=0位4-1=0

位4=卤1

So the complex eigenvalues is位1,2=卤1,位3,4=卤1

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Most popular questions from this chapter

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.

if A is a 2脳2matrix with t r A = 5and det A = - 14what are the eigenvalues of A?

True or false? If the determinant of a 2 脳 2 matrix A is negative, then A has two distinct real eigenvalues.

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 through20,find all (real)eigenvalues.Then find a basis of each eigenspaces,and diagonalize A, if you can. Do not use technology.

[1111]

Find a 2脳2matrixsuch that

X鈫�(t)=[2t-6t2t+6t]is a trajectory of the dynamical systemrole="math" localid="1659527385729" x鈫�(t+1)=Ax鈫�(t)
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