MARTINA BALAGOVIC: Hi.
Welcome to recitation.
Today's problem is about positive definite matrices.
And it's asking us: for which values of the parameter
c, which is sitting here in the matrix, is the matrix B-- 2,
minus 1, minus 1; minus 1, 2, minus 1; minus 1, minus 1, 2
plus c-- positive definite, and for which values of c
is it positive semidefinite?
I'm going to leave you alone with the problem.
You should pause the video and then
come back and compare your solution with mine.
And we're back.
As you remember from the lectures,
there are several tests that you can do on matrices to find out
if they're positive definite and if they're
positive semidefinite.
And I'm going to demonstrate three to you.
First, I'm going to do the one that you should do in case
you have very little time and you're
asked to do a problem like this on the test, which is
of course the determinant test.
The determinant test asks us to compute
determinants of the matrices in the upper left corner
of all sizes.
And it says that it's going to be positive definite
if they're all greater than 0 and positive semidefinite
if some 0's sneak into that sequence.
So let's calculate the determinants.
The first determinant is the determinant of this tiny matrix
here.
So it's just 2.
The next one is the determinant of this two by two submatrix,
2, minus 1; minus 1, 2; which is equal to 4 minus 1, which is 3.
And finally, we have the determinant
of B, which I'm going to calculate for you.
I'm going to calculate it, I'm going
to decompose it along the first line, first row.
So it's 2 times the determinant of this submatrix, 2, minus 1;
minus 1, 2 plus c; minus minus 1, this one here,
times this determinant, which is minus 1, minus 1;
minus 1, 2 plus c.
And then plus minus 1, this one here, times this determinant,
which is minus 1, 2; minus 1, minus 1.
And in total, that's 2 times 4 plus 2c minus 1 plus minus 2
minus c minus 1 and minus 1 plus 2.
And in total, this should give us
6 plus 4c minus 3 minus c minus 3, which is 3c.
So let's look at the determinants.
2 is positive.
3 is positive.
3c is positive.
So the answer is: the matrix is positive definite
if c is bigger than 0, and it's positive semidefinite
if it's either strictly bigger than 0 or equal to 0.
And that's all.
If you're on the test, this is everything that you should do.
Now let me show you two more tests
to demonstrate that first, they take longer, and second, to see
these numbers and their quotients show up
in other tests and to try to convince you
that these tests really are equivalent.
Let me do the pivots test for you.
So we take our matrix B, 2, minus 1, minus 1; minus 1, 2,
minus 1; minus 1, minus 1, 2 plus c.
And let's pretend we're solving a system that
has this as a matrix.
So we multiply the first row by 1/2,
and we add it to the second and to the third row.
We get 2, minus 1, minus 1; 0, 3/2, minus 3/2; 0, minus 3/2,
3/2 plus c.
So the first column is good.
And then we just replace the third column
with the third column plus the second column.
And we get 2, minus 1, minus 1; 0, 3/2, minus 3/2; 0, 0, and c.
And so the pivots are 2, 3/2, and c.
And again, the answer is as before.
It's positive definite if c is strictly bigger than 0,
and it's positive semidefinite if c is greater or equal to 0.
But I want you to notice something else here.
So before, we had 2, 3, and 3c.
And now for determinants, as these pivots,
we have 2, 3/2, which is the second determinant
over the first determinant, and c,
which can be thought of as 3c over 3,
so the third determinant over the second determinant.
And something like this is always going to happen.
And finally, let me do the energy test, or completing
the square.
So one of the definitions of positive definite,
one could say the definition because
of which we are really interested in such matrices,
is the following.
It's positive definite if, when we consider
this quadratic form, so a form that maps
x, y, and z to this expression here that's
going to be quadratic in x, y, and z, it's positive
semidefinite if this is always greater or equal than 0.
And it's positive definite if, when
we have an expression like this and try to solve this equals 0,
the only solution is that x, y, and z all have to be 0.
So let's try calculating this form,
completing the squares on it, and seeing these numbers
show up again.
So when I multiply this like this, put in a B,
do the multiplication, we get something like this.
2 x squared plus 2 y squared plus 2 plus c z squared minus
2x*y minus 2x*z minus 2y*z.
And now let's try completing the squares using the formulas
that I prepared for you up here in this pop-up window
in the corner.
So let's try completing-- we have
a formula for the square of a plus
b plus c and the square of a plus b.
So first, we try to get something squared so that this
something has all the x's that appear here,
so that we get something squared plus some expression that only
takes y's and z's.
I'm not going to do the calculation in front of you
to further embarrass myself with it.
But let me tell you that what you get
is 2 times x minus 1/2 y minus 1/2 z squared.
And this ate all the x's that showed up here.
The remainder only has y's and z's.
When you use the second of those formulas in a pop-up window
to complete the square of y's and z's, you
get 3/2 times y minus z squared.
So this took up all the y's, and we're
left just with a z that comes as c times z squared.
And now, let's look at the following.
This is a square of some real number.
So that's positive.
This is a square of some real number, so that's positive.
And this is a square of some real number,
so that's positive.
They're all multiplied by positive numbers, 2, 3/2,
and c, which we've already seen here.
And so if c is bigger or equal than 0,
this is certainly always bigger or equal than 0.
Now to the question of if this can be equal to 0 without x, y,
and z all being 0, well, let's look at two cases.
If c is strictly bigger than 0, then
let me write this matrix here as 2, 3/2 and c times 1,
minus 1/2, minus 1/2; 0, 1, minus 1; and 0, 0, 1.
Let's imagine c is strictly bigger than 0.
And let's see when can this expression be equal to 0.
Well, as we said, it's a sum of certain squares.
They're all greater or equal to 0,
so they all have to be equal to 0 for this expression to be 0.
In other words, z needs to be equal to 0.
y minus z needs to be equal to 0.
And x minus 1/2 y minus 1/2 z also needs to be equal to 0.
And since this matrix has all the pivots,
this can only happen if x, y, and z are all equal to 0.
On the other hand, if c is equal to 0,
then let me write this matrix here again.
Take 2 out here.
Take 3/2 out here.
And write again 1, minus 1/2, minus 1/2; 0, 1, minus 1;
and 0, 0, 0.
And so imagining that c is 0, when can this whole expression
be equal to 0?
Well, the last bit is already 0.
We need to have y minus z equal to 0.
And we need to have 1 times x minus 1/2 times y minus 1/2
times z also equal to 0.
So we need to have x, y, z in the kernel of this matrix.
But this matrix only has two pivots.
It has one free variable.
So we can find a solution to this times x,
y, z equals 0 that's not 0 itself.
Namely, if you remember how to solve systems like this,
we set z equal to 1.
From this, we calculate that y has to be 1.
And then from this, we calculate that x has to be 1.
And you can check that for in case c is 0, this thing
here, when you plug it in here, you really get 0.
In fact, this thing here is in the kernel
of the matrix B. In fact, in case c
is 0, the columns of matrix B sum up to 0 because of this
here.
And that's all I wanted to say for today.