Thread: Find the Eigenvalues and Eigenspaces

Is my working correct?

Question: Find the eigenvalues and the corresponding eigenspaces for matrix A:

A=
6 -4
3 -1

|A - lambda * I| = 0

6-lambda -4
3 -1-lambda

(6-lambda)(-1-lambda) - (3)(-4)

-6 -6*lambda + lambda + lambda^2 + 12

6 -5*lambda + lambda^2

Factorize 6 -5*lambda + lambda^2

(lambda - 3)(lambda - 2)

Eigenvalues of A:

lambda = 3
lambda = 2

E(3) =

6x - 4y = 3x
3x - y = 3y

-4y = -3x
3x = 4y

Span = ?

The eigenspaces are ?

E(2) =

6x - 4y = 2x
3x - y = 2y

-4y = -4x
3x = 3y

Span = ?

The eigenspaces are ?

Where did I go wrong? How do I find the eigenvalues, what is the "span", and how do I find it?

Your work so far is just fine, I would say. You are a bit confused in finding the eigenvectors (they span the eigenspace). Solve the system for Since you have chosen such that the matrix is singular, you're guaranteed that the eigenvectors you find are not unique. In fact, any scalar multiple of an eigenvector is an eigenvector. So what do you get when you solve the system for

Thanks for your reply Ackbeet.

Here is what I got so far...

(A - lambda I)x = 0

6-lambda -4
3 -1-lambda

Originally Posted by sparky

Eigenvalues of A:

lambda = 3
lambda = 2

E(3) =

6x - 4y = 3x
3x - y = 3y

-4y = -3x
3x = 4y

Span = ?

The eigenspaces are ?

E(2) =

6x - 4y = 2x
3x - y = 2y

-4y = -4x
3x = 3y

Span = ?

The eigenspaces are ?

Where did I go wrong? How do I find the eigenvalues, what is the "span", and how do I find it?

I find it easier to do:









The eigenvector for lamba = 3 is



Do the same for lambda = 2. Then your eigenspace will be