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Chapter 7: Q8E (page 336)

For each of the matrices in Exercises 1 through 13, find all real eigenvalues, with their algebraic multiplicities. Show your work. Do not use technology.

[-1-1-1-1-1-1-1-1-1]

Short Answer

Expert verified

Eigenvalues are:

位1,2=0,almu0=2位3=3,almu3=3

Step by step solution

01

Eigenvalues

  • In linear algebra, an eigenvector or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted by 位, is the factor by which the eigenvector is scaled.
  • Eigenvalues of a triangular matrix are its diagonal matrix.
02

Step 2: Finding all real eigenvalues, with their algebraic multiplicities: 

We can clearly see that,

detA-位=0-1-位-1-1-1-1-位-1-1-1-1-位=01-位3-1-1-3-1-位=0位2位-3=0位1,2=0,位3=3

Hence, the answer is:

位1,2=0,almu0=2位3=3,almu0=3

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Most popular questions from this chapter

Find an eigenbasis of given matrix and diagonalize it.

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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

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Two interacting populations of coyotes and roadrunners can be modeled by the recursive equations

h(t + 1) = 4h(t)-2f(t)

f(t + 1) = h(t) + f(t).

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(a)h(0)=f(0)=100(b)h(0)200,f(0)=100(c)h(0)600,f(0)=500
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