Linear Algebra: Eigenvectors and Eigenspaces for a 3x3 matrix

In the last video we set out to
find the eigenvalues values

of this 3 by 3 matrix, A.

And we said, look an eigenvalue
is any value,

lambda, that satisfies
this equation if v

is a non-zero vector.

And that says, any value,
lambda, that satisfies this

equation for v is a
non-zero vector.

Then we just did a little bit
of I guess we could call it

vector algebra up here
to come up with that.

You can review that
video if you like.

And then we determined, look
the only way that this is

going to have a non-zero
solution is if this matrix has

a non-trivial null space.

And only non-invertible matrices
have a non-trivial

null space.

Or, only matrices that have
a determinant of 0 have

non-trivial null spaces.

So you do that, you got your
characteristic polynomial, and

we were able to solve it.

And we got our eigenvalues where
lambda is equal to 3 and

lambda is equal to minus 3.

So now, let's do-- what I
consider the more interesting

part-- is actually find out
the eigenvectors or the

eigenspaces.

So we can go back to this
equation, for any eigenvalue

this must be true.

This must be true but this
is easier to work with.

And so, this matrix right here
times your eigenvector must be

equal 0 for any given
eigenvalue.

This matrix right here--
I've just copied

and pasted from above.

I marked it up with the Rule
of Sarrus so you can ignore

those lines-- is just
this matrix right

here for any lambda.

Lambda times the identity
matrix minus A

ends up being this.

So let's take this matrix for
each of our lambdas and then

solve for our eigenvectors
or our eigenspaces.

So let me take the case of
lambda is equal to 3 first. So

if lambda is equal to 3, this
matrix becomes lambda plus 1

is 4, lambda minus 2 is 1,
lambda minus 2 is 1.

And then all of the other terms
stay the same, minus 2,

minus 2, minus 2, 1,
minus 2 and 1.

And then this times that vector,
v, or our eigenvector

v is equal to 0.

Or we could say that the
eigenspace for the eigenvalue

3 is the null space
of this matrix.

Which is not this matrix.

It's lambda times the
identity minus A.

So the null space of this matrix
is the eigenspace.

So all of the values that
satisfy this make up the

eigenvectors of the eigenspace
of lambda is equal to 3.

So let's just solve for this.

So the null space of this guy--
we could just put in

reduced row echelon form-- the
null space of this guy is the

same thing as the null space
of this guy in reduced row

echelon form.

So let's put this in reduced
row echelon form.

So the first thing I
want to do-- let me

just do it down here.

So let me-- I'll keep my first
row the same for now.

4 minus 2, minus 2.

And let me replace my second row
with my second row times 2

plus my first row.

So minus 2 times
2 plus 1 is 0.

1 times 2 plus minus 2 is 0.

1 times 2 plus minus 2 is 0.

This row is the same
as this row.

So I'm going to do
the same thing.

Minus 2 times 2 plus 4 is 0.

1 times 2 plus 2 is 0.

And then 1 times 2 plus
minus 2 is 0.

So the solutions to this
equation are the same as the

solutions to this equation.

Let me write it like this.

Instead of just writing
the vector, v,

let me write it out.

So v1, v2, v3 are going to
be equal to the 0 vector.

0, 0.

Just rewriting it slightly
different.

And so these two rows, or these
two equations, give us

no information.

The only one is this row up
here, which tells us that 4

times v1 minus 2 times v2--
actually this wasn't complete

reduced row echelon form
but close enough.

It's easy for us to work with--
4 times v1 minus 2

times v2 minus 2 times
v3 is equal to 0.

Let's just divide by 4.

I could've just divided by 4
here, which might have made it

skipped a step.

But if you divide by 4 you get
v1 minus 1/2 v2 minus 1/2 v3

is equal to 0.

Or, v1 is equal to 1/2
v2 plus 1/2 v3.

Just added these guys to both
sides of the equation.

Or we could say, let's say that
v2 is equal to-- yeah I

don't know, I'm going to just
put some random number-- a,

and v3 is equal to b, then we
can say-- and then v1 would be

equal to 1/2 a plus 1/2 b.

We can say that the eigenspace
for lambda is equal to 3, is

the set of all of vectors, v1,
v2, v3, that are equal to a

times times-- v2 is a, right?

So v2 is equal to a times 1.

v3 has no a in it.

So it's a times 0.

Plus b times-- v2 is just a.

v2 has no b in it.

So it's 0.

v3 is 1 times-- so 0 times
a plus 1 times b.

And then v1 is 1/2
a plus 1/2 b.

For any a and b, such
that a and b are

members of the reals.

Just to be a little bit
formal about it.

So that's our-- any vector
that satisfies this is an

eigenvector.

And they're the eigenvectors
that correspond to eigenvalue

lambda is equal to 3.

So if you apply the matrix
transformation to any of these

vectors, you're just going
to scale them up by 3.

Let me write this way.

The eigenspace for lambda is
equal to 3, is equal to the

span, all of the potential
linear combinations of this

guy and that guy.

So 1/2, 1, 0.

And 1/2, 0, 1.

So that's only one of
the eigenspaces.

That's the one that
corresponds to

lambda is equal to 3.

Let's do the one that
corresponds to lambda is equal

to minus 3.

So if lambda is equal to minus
3-- I'll do it up here, I

think I have enough space--
lambda is equal to minus 3.

This matrix becomes-- I'll do
the diagonals-- minus 3 plus 1

is minus 2.

Minus 3 minus 2 is minus 5.

Minus 3 minus 2 is minus 5.

And all the other things
don't change.

Minus 2, minus 2, 1.

Minus 2, minus 2 and 1.

And then that times vectors
in the eigenspace that

corresponds to lambda is equal
to minus 3, is going to be

equal to 0.

I'm just applying this equation
right here which we

just derived from that
one over there.

So, the eigenspace that
corresponds to lambda is equal

to minus 3, is the null space,
this matrix right here, are

all the vectors that satisfy
this equation.

So what is-- the null space of
this is the same thing as the

null space of this in reduced
row echelon form So let's put

it in reduced row
echelon form.

So the first thing I want to do,
I'm going to keep my first

row the same.

I'm going to write a little bit
smaller than I normally do

because I think I'm going
to run out of space.

So minus 2, minus 2, minus 2.

Actually let me just
do it this way.

I will skip some steps.

Let's just divide the first
row by minus 2.

So we get 1, 1, 1.

And then let's replace this
second row with the second row

plus this version of
the first row.

So this guy plus that guy is 0
minus 5 plus minus-- or let me

say this way.

Let me replace it with
the first row

minus the second row.

So minus 2 minus minus 2 is 0.

Minus 2 minus minus
5 is plus 3.

And then minus 2 minus
1 is minus 3.

And then let me do
the last row in a

different color for fun.

And I'll do the same thing.

I'll do this row
minus this row.

So minus 2 minus
minus 2 is a 0.

Minus 2 plus 2.

Minus 2 minus 1 is minus 3.

And then we have minus
2 minus minus 5.

So it's minus 2 plus 5.

So that is 3.

Now let me replace-- and I'll
do it in two steps.

So this is 1, 1, 1.

I'll just keep it like that.

And actually, well let me
just keep it like that.

And then let me replace my third
row with my third row

plus my second row.

It'll just zero out.

If you add these terms, these
all just become 0.

That guy got zeroed out.

And let me take my second
row and divide it by 3.

So this becomes 0, 1, minus 1.

And I'm almost there.

I'll do it in orange.

So let me replace my first row
with my first row minus my

second row.

So this becomes 1, 0, and then
1 minus minus 1 is 2.

1 minus minus 1 is 2.

And then in the second
row is 0, 1, minus 1.

And then the last
row is 0, 0, 0.

So any v that satisfies this
equation will also

satisfy this guy.

This guy's null space is going
to be the null space of that

guy in reduced row
echelon form.

So v1, v2, v3 is equal
to 0, 0, 0.

Let me move this.

Because I've officially
run out of space.

So let me move this lower
down where I have

some free real estate.

Let me move it down here.

This corresponds to lambda
is equal to minus 3.

This was lambda is equal to
minus 3, just to make us--

it's not related to this
stuff right here.

So what are all of the v1s, v2s
and v3s that satisfy this?

So if we say that v3
is equal to t.

If v3 is equal to t, then
what do we have here?

We have-- this tells us that
v2 minus v3 is equal to 0.

So that tells us that v2 minus
v3-- 0 times v1 plus v2 minus

v3 is equal to 0.

Or that v2 is equal to v3,
which is equal to t.

That's what that second
equation tells us.

And then the third equation
tells us, or the top equation

tells us, v1 times 1-- so v1
plus 0 times v2 plus 2 times

v3 is equal to 0.

Or v1 is equal to minus 2v3 is
equal to minus 2 times t.

So the eigenspace that
corresponds to lambda is equal

to minus 3 is equal to the set
of all the vectors, v1, v2 and

v3, where-- well, it's equal
to t times-- v3 is just t.

v3 was just t.

v2 also just ends up being t.

So 1 times t.

And v1 is minus 2 times t.

For t is any real number.

Or another way to say it is that
the eigenspace for lambda

is equal to minus 3 is equal
to the span-- I wrote this

really messy-- where lambda is
equal to minus 3 is equal to

the span of the vector
minus 2, 1, and 1.

Just like that.

It looks interesting.

Because if you take this guy
and dot it with either of

these guys, I think you get 0.

Is that definitely the case?

Take minus 2 times 1/2, you
get a minus 1 there.

Then you have a plus 1.

That's 0.

And then minus 2 times 1/2.

Yeah.

You dot it with either of
these guys you get 0.

So this line is orthogonal
to that plane.

Very interesting.

So let's just graph it just so
we have a good visualization

of what we're doing.

So we had that 3
by 3 matrix, A.

It represents some
transformation in R3.

And it has two eigenvalues.

And each of those have a
corresponding eigenspace.

So the eigenspace that
corresponds to the eigenvalue

3 is a plane in R3.

So this is the eigenspace for
lambda is equal to 3.

And it's the span of these
two vectors right there.

So if I draw them, maybe
they're like that.

Just like that.

And then the eigenspace
for lambda is equal to

minus 3 is a line.

It's a line that's perpendicular
to this plane.

It's a line like that.

It's the span of this guy.

Maybe if I draw that vector,
that vector might look

something like this.

And it's the span of that guy.

So what this tells us, this is
the eigenspace for lambda is

equal to minus 3.

So what that tells us-- just
to make sure we are

interpreting our eigenvalues and
eigenspaces correctly-- is

look, you give me any
eigenvector, you give me any

vector in this, you give me any
vector right here, let's

say that is vector x.

If I apply the transformation,
if I multiply it it by a, I'm

going to have 3 times that.

Because it's in the eigenspace
where lambda is equal to 3.

So if I were to apply a times
x, a times x would be just 3

times that.

So that would be a times x.

That's what it tells me.

This would be true for
any of these guys.

If this was x, and you took a
times x, it's going to be 3

times as long.

Now these guys over here, if you
have some vector in this

eigenspace that corresponds to
lambda is equal to 3, and you

apply the transformation.

Let's say that this
is x right there.

If you took the transformation
of x, it's going to make it 3

times longer in the opposite
direction.

It's still going to
be on this line.

So it's going to go
down like this.

And that would be a times x.

It would be the same, it'd be
3 times this length, but in

the opposite direction.

Because it corresponds to lambda
is equal to minus 3.

So anyway, we've, I think,
made a great achievement.

We've not only figured out the
eigenvalues for a 3 by 3

matrix, we now have figured out
all of the eigenvectors.

Which are-- there's an infinite
number-- but they

represent 2 eigenspaces that
correspond to those two

eigenvalues, or minus 3 and 3.

See you in the next video.