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Chapter 7: Q28E (page 372)

Suppose a 3 脳 3 matrix A has the real eigenvalue 2 and two complex conjugate eigenvalues. Also, suppose that det A = 50 and tr A = 8. Find the complex eigenvalues.

Short Answer

Expert verified

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

Step by step solution

01

Define eigenvalue

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

02

Find the complex eigenvalues

Lets consider,

位1=2,位2=a+bi,位3=a-bi

Using theorem 4.5.5,

We will get,2(a2+b2)=50 and 2a+2=8.

Solve2a+2=8 for a,

2a=6a=3

Substitute a=3 into2a2+b2=50 and solve for b.

232+b2=5029+b2=509+b2=25

b2=16b=卤4

So, we get the solution as role="math" localid="1659605249127" 位2=3卤4i.

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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.

find an eigenbasis for the given matrice and diagonalize:

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For a given eigenvalue, find a basis of the associated eigenspace. Use the geometric multiplicities of the eigenvalues to determine whether a matrix is diagonalizable. For each of the matrices A in Exercises 1 through 20, find all (real) eigenvalues. Then find a basis of each eigenspace, and diagonalize A, if you can. Do not use technology

8((110022003)

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