Purchase Solution

Matrices Eigenvalues, eigenspace, and eigenbasis.

Not what you're looking for?

Ask Custom Question

Matrices Eigenvalues, eigenspace, and eigenbasis.

See attached file.

For each of the matrices, find all (real) eigenvalues. Then find a basis of each eigenspace, and find an eigenbasis, if you can.
[■(7&8@0&9)]
[■(1&1@1&1)]
[■(6&3@2&7)]
[■(0&-1@1&2)]
[■(1&1&0@0&2&2@0&0&3)]
[■(1&1&0@0&1&1@0&0&1)]
[■(■(0&0@0&1)&■(0&0@0&1)@■(0&0@0&0)&■(0&0@0&1))]

Attachments
Purchase this Solution

Solution Summary

Matrices eigenvalues, eigenspaces and eigenbasis is examined.

Solution Preview

1.
The matrix is:
(1.1)
This is an upper triangular matrix, hence the eigenvalues are the elements of the diagonal:
(1.2)
The first eigenvector:

(1.3)
The second eigenvector:

(1.4)
We have two distinct eigenvalues and two distinct eigenvectors. The algebraic multiplicity (m=1) of each of the eigenvalues equal their respective geometric multiplicity, hence these two eigenvectors form an eigenbasis.

2.
The matrix is:
(2.1)
The eigenvalues are:

(2.2)
The first eigenvector:

Therefore:
(2.3)
The second eigenvector:

(2.4)
As before we have two distinct eigenvalues and two distinct eigenvectors. The algebraic multiplicity (m=1) of each of the eigenvalues equal their respective geometric multiplicity, hence these two eigenvectors form an eigenbasis. ...

Purchase this Solution


Free BrainMass Quizzes
Graphs and Functions

This quiz helps you easily identify a function and test your understanding of ranges, domains , function inverses and transformations.

Multiplying Complex Numbers

This is a short quiz to check your understanding of multiplication of complex numbers in rectangular form.

Know Your Linear Equations

Each question is a choice-summary multiple choice question that will present you with a linear equation and then make 4 statements about that equation. You must determine which of the 4 statements are true (if any) in regards to the equation.

Solving quadratic inequalities

This quiz test you on how well you are familiar with solving quadratic inequalities.

Exponential Expressions

In this quiz, you will have a chance to practice basic terminology of exponential expressions and how to evaluate them.