How do eigenvalues and eigenvectors relate to matrices?

[S_Flaherty]

I have to be able to figure out eigenvalues and eigenvectors for 2x2 and 3x3 matrices for my physics course, but I have never taken linear algebra so I obviously have no idea what they even are. I need someone to basically teach me how to solve these problems because I have no knowledge of this material and cannot find any useful source to help me.

First I have to find the eigenvalues and vectors for

[a 0 1]
[0 b 0]
[0 0 c] where a =/= b =/= c.

I looked around online and could only find that the answers for the eigenvalues are a, b, and c but I have no idea how that answer is derived. And I have no idea how to solve for eigenvectors except so any help would be great.

Next I have to do the same thing for

[a a 0]
[2a 0 0]
[0 a a] where a =/= 0.

Again I know that the solution for the values is -a, a, and 2a but no clue on how to actually do the problem.

I know nothing of this material except how to figure out the determinant of a matrix. So detailed steps on how to solve for eigenvalues and eigenvectors would be greatly appreciated.

 

[I like Serena]

An eigenvalue λ and its eigenvector v are defined by:

Av = λv

That is, you try to find a vector that if you multiply A with it you find the same vector, except for length.

To solve this, you do:
Av - λv = 0
(A - λI)v = 0

This means (A - λI) must be singular, so its determinant must be zero.
det(A - λI)=0 yields a polynomial in λ, that is called the "characteristic polynomial".
Its roots are the eigenvalues of A.

Once you have an eigenvalue, you solve:
(A - λI)v = 0
to find a vector v (or its multiple) that satisfies the equation.

 

[I like Serena]

Applied to your first matrix you have:
det(A - λI)=0
(a-λ)(b-λ)(c-λ)=0
Roots are a, b, and c.

For the first eigenvalue "a", you get:
A - aI =
[0 0 1]
[0 b 0]
[0 0 c]

The vector v=(1,0,0) satisfies (A - aI)v = 0, meaning that is the eigenvector.

 

[S_Flaherty]

Applied to your first matrix you have:
det(A - λI)=0
(a-λ)(b-λ)(c-λ)=0
Roots are a, b, and c.

For the first eigenvalue "a", you get:
A - aI =
[0 0 1]
[0 b 0]
[0 0 c]

The vector v=(1,0,0) satisfies (A - aI)v = 0, meaning that is the eigenvector.

So i understand how you get the eigenvalues but I'm still confused with how you are arriving at the answer for the eigenvector.

 

[Mark44]

The matrix A - aI can be row reduced to
$$\begin{bmatrix}0&0&1\\0&1&0\\0&0&0 \end{bmatrix}$$

This represents the equation (A - aI)x = 0, where x is a column vector with coordinates x₁, x₂ and x₃.

The reduced matrix represents the system
x₃ = 0
x₂ = 0
with x₁ being arbitrary. Since it's arbitrary, it's reasonable to set it to 1, giving the vector <1, 0, 0> as an eigenvector for the eigenvalue a.