Eigenvalues And Eigenvectors

Question 1

Eigenvalue

Answer 1

Given
Av = λ*v (where λ is a real number constant)
lambda is the eigen value

in other words, a constant that if multiplied by a special input vector the output made by that input is a that constant multiplied by the input vector

Question 2

Eigen

Answer 2

means “self” in german

Question 3

Eigenvector

Answer 3

A vector v where
Av = const*v

Question 4

How to find the eigenvalues

Answer 4

Solve det(A - λ*I) = 0 for lambda.

Question 5

Characteristic polynomial of a linear transformation A

Answer 5

Pₐ(λ) = det(A- λ*I)
The lambdas for which it is zero are defined as the eigenvalues of A.

Question 6

Eigenspace

Answer 6

Vλ(A) = the set of all v in domain for which Av = lambda*v for some lambda.

Question 7

How to find the basis of the eigenspace of a linear transformation A

Answer 7

1. Find the eigenvalues by solving dim(A - λ*I) = 0 for all lambda.
2. For each eigenvalue, plug into
   (A - λI)v = 0 and solve for v. (This is simply a homogenous system, we solve it like any other.)