.

If I'm given a square matrix 'A'

an eigenvector associated with that matrix is a vector

so that when I multiply the vector by 'A'

the effect of multiplication is a scaling

of that eigenvector

and the scaling value is called the eigenvalue

So if I'm given a square matrix 'A'

an eigenvector, we'll call it 'v'

associated with that matrix

is a vector so that, when I multiply on the left

by the matrix 'A'

the effect of that multiplication is a scaling

of the eigenvector by this value 'lambda'

which is the eigenvalue associated

with that eigenvector

So, one more time

The whole idea behind eigenvalues and eigenvectors

is that, given a matrix 'A' - a square matrix

an eigenvector associated with that matrix

is a vector such that when I multiply

by 'A' on the left,

the effect of the multiplication is a scaling

of the vector by its corresponding eigenvalue

The eigenvector 'v' by definition is not the zero vector

Let's suppose we have a nice matrix

Make things small in dimension here

let's say 3 by 1 and 1 by 3

Now I want to take that matrix and

we'll define a vector

this will turn out to be an eigenvector

as we'll see in just a moment

Let's say a vector (1,1)

So what is the result of multiplication here?

'A' on the left times 'v'

So multiply the matrix [(3,1) (1,3)]

times the vector(1,1)

Notice this matrix multiplication is defined again

a vector can be considered as something of a

degenerate or one dimensional matrix

So when I perform the matrix multiplication

the dot product here (3,1) dotted with (1,1)

results in 4

(1,3) similarly dotted with (1,1)

results in 4

So what do we have here?

We have Av, my matrix times this vector

Well the action of that multiplication

in a geometric sense

is a scaling of the original vector 'v'

by a value of 4

So Av in other words here is equal to

lambda, where lambda is 4, my scaling factor

times the original matrix

So the bottom line is, to be eigenvector,

when I multiply on the left by my matrix

the action of this multiplication

results in a scaling of my eigenvector

in this case by the value 4

I'd like to shed a little more light

on this idea of eigenvalues and eigenvectors

now from a geometric perspective

So we'll continue with the example

where we called the matrix 'A'

this 2 by 2 matrix, [(3,1) (1,3)]

and as you'll recall from the earlier example

One of the eigenvectors associated

with that matrix

We'll call that vector v_1

is the vector (1,1)

So I've drawn that now on the plane

And then the resultant action

of multiplying the vector v_1 on the left

by the matrix aforementioned 'A' here

was this stretching or scaling by a factor 4 here

And that was the lambda associated with

that eigenvector

So we can think of this geometrically here

'Av_1' results in 4 times v_1

Okay, so there's a nice kind of geometric rendering

of what it means to be an eigenvalue

and an eigenvector for a matrix

As I also said earlier

this particular matrix, as is common,

with lots of matrices

has other eigenvectors associated with it

And I'll show you how to compute some of those

things in just a moment here

But one of the other eigenvectors

associated with that particular matrix

We'll call that vector v_2

is the vector as I've drawn here (-1,1)

And the eigenvalue associated with that

particular eigenvector is the value 2

So in other words, lambda is 2

for this particular eigenvector

So the result then, of multiplying

let's say the vector v_2 on the left

by the matrix A

is that I get this scaled version

in other words twice times the original vector here