.
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