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Chapter 7: Q5E (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.

[11-156-7]

Short Answer

Expert verified

No real eigenvalues.

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

Find the characteristic polynomial as:

det(A-位滨n)=det11-位-156-7-位=11-位-7-位--15.6=位2-4位-77+90=位2-4位+13

Eigenvalues are:

位=2+3iwith multiplicity of 1

位=2-3iwith multiplicity of 1

Hence, there is no real eignevalues, as the obtained values are complex.

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Most popular questions from this chapter

Consider the matrix A=|abba|where aand bare arbitrary constants. Find all eigenvalues of A.

Question: If a vectorv鈫�is an eigenvector of both AandB, isv鈫� necessarily an eigenvector ofAB?

For m鈮�n, find the dimension of the space of alln脳nmatricesfor which all the vectorse鈬赌1,....,e鈬赌mare eigenvectors.

Find an eigenbasis for the given matrice and diagonalize:

A=[1111]

24: Find all eigenvalues of the positive transition matrix

A=[0.50.250.50.75]

See Definitions 2.1.4 and 2.3.10.
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