Understanding how to find the Algebraic/Geometric multiplicity of an eigenvalue

Question

I'm currently in an introductory course to linear algebra and i'm confused as to what geometric multiplicity is and how to find it... I've been looking at my textbook's example for a bit, and I understand how everything is done until the highlighted area.

Link of the example: http://puu.sh/A5SrC/612bf800d2.jpg

I don't get why they are picking values for s and r, and how the algebraic and geometric multiplicity are both 2 for the eigenvalue 2 shown in the example...

edit: the algebraic/geometric multiplicity of 2 is shown later in the textbook, which is not in the picture shown above.

Answer 1

I'm in an introductory course too! So let me have a crack at this.

Let $\lambda_0$ as an eigenvalue of some linear transformation $T$ and $(\lambda - \lambda_0)^k$ be the highest power of $\lambda - \lambda_0$ that divides the characteristic polynomial, $p_T(\lambda)$. Then $k$ is said to be the algebraic multiplicity.

This is the definition for algebraic multiplicity I've been taught, what it means is, basically, how many times does an eigenvalue appear. So when you investigate your characteristic polynomial and conclude what the eigenvalues are, how many times does each eigenvalue come up?

The geometric multiplicity of the eigenvalue $\lambda_0$ is the dimension of its eigenspace, i.e., $dim {{E_\lambda}_0}$. It is the maximal number of linear independant eigenvectors belonging to $\lambda_0$.

This is the definition for geometric multiplicity I've been taught, this one is a little bit tricker.

Let me explain (at least the method) like this, when we have found an eigenvalue, $\lambda_0$, we can continue to find their associated eigenvectors by evaluating;

$$null(A-\lambda_0I)$$

(I will link an awesome video series that will get you familiar as to really why this is the case). For now, you should understand that this will give you a set of restrictions from of which you derive your associated eigenvector from.

In your case (as the linked photo shows), we have determined such a matrix (using $\lambda_0 = 1$) from the equation above, row echeon reduced it and stumbled across a nice looking restriction.

The restriction is $x_2-x_3 = 0$, recall we're finding vectors in the form of $(x_1, x_2, x_3)^T$. So how do we apply this restriction you may ask?

Well it's saying, $x_2=x_3$ and there is no restriction on $x_1$, hence indeed we have a eigenvector specification of the form $(s,r,r)^T$ for $r,s$ scalars. So from here, list all the different types of linearly independent vectors that could possibly create. And boom, you've got your eigenvectors.

Referring back to the definition of geometric multiplicity, count how many linearly independant eigenvectors you have belonging to $\lambda_0$ and that'll be it's respective geometric multiplicity.

Hope this helped!

I reckon you should check this out too: https://www.khanacademy.org/math/linear-algebra/alternate-bases/eigen-everything/v/linear-algebra-introduction-to-eigenvalues-and-eigenvectors