Eigen-everything
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Introduction to Eigenvalues and Eigenvectors
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Proof of formula for determining Eigenvalues
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Example solving for the eigenvalues of a 2x2 matrix
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Finding Eigenvectors and Eigenspaces example
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Eigenvalues of a 3x3 matrix
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Eigenvectors and Eigenspaces for a 3x3 matrix
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Showing that an eigenbasis makes for good coordinate systems
Finding Eigenvectors and Eigenspaces example Finding the eigenvectors and eigenspaces of a 2x2 matrix
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- In the last video, we started with the 2 by 2 matrix A is
- equal to 1, 2, 4, 3.
- And we used the fact that lambda is an eigenvalue of A,
- if and only if, the determinate of lambda times
- the identity matrix-- in this case it's a 2 by 2 identity
- matrix-- minus A is equal to 0.
- This gave us a characteristic polynomial and we solved for
- that and we said, well, the eigenvalues for A are lambda
- is equal to 5 and lambda is equal to negative 1.
- That's what we saw in the last video.
- We said that if you were trying to solve A times some
- eigenvector is equal to lambda times that eigenvector, the
- two lambdas, which this equation can be solved for,
- are the lambdas 5 and minus 1.
- Assuming nonzero eigenvectors.
- So we have our eigenvalues, but I don't even call that
- half the battle.
- What we really want is our eigenvectors and our
- eigenvalues.
- So let's see if we can do that.
- So if we manipulate this equation a little bit and
- we've manipulate it in the past. Actually, we've even
- come up with this statement over here.
- We can rewrite this over here as the 0 vector is equal to
- lambda times my eigenvector minus A times my eigenvector.
- I just subtracted Av from both sides.
- We know lambda times some eigenvector is the same thing
- as lambda times the identity matrix times that eigenvector.
- So all I'm doing is rewriting this like that.
- You multiply the identity matrix times an eigenvector or
- times any vector, you're just going to get that vector.
- So these two things are equivalent.
- Minus Av.
- That's still going to be able to the 0 vector.
- So far all I've done is manipulated this thing.
- This is really how we got to that thing up there.
- You factor out the v so to speak because we know that
- matrix vector products exhibit the distributive property.
- And we get lambda times the identity matrix minus A times
- my eigenvector have got to be equal to 0.
- Or another way to say it is, for any lambda eigenvalue, and
- let me write it for any eigenvalue lambda, the
- eigenvectors that correspond to that lambda, we can call
- that the eigenspace for a lambda.
- So that's a new word, eigenspace.
- Eigenspace just means all of the eigenvectors that
- correspond to some eigenvalue.
- The eigenspace for some particular eigenvalue is going
- to be equal to the set of vectors that
- satisfy this equation.
- Well, the set of vectors that satisfy this equation is just
- the null space of that right there.
- So it's equal to the null space of this
- matrix right there.
- The null space of lambda times the identity matrix.
- And by an identity matrix minus A.
- And so everything I've done here, this is true-- this is
- the general case.
- But now we can apply this notion to this
- matrix A right here.
- So we know that 5 is an eigenvalue.
- Let's say for lambda is equal to 5, the eigenspace that
- corresponds to 5 is equal to the null space of?
- Well, what is 5 times the identity matrix?
- It's going to be the 2 by 2 identity matrix.
- 5 times the identity matrix is just 5, 0, 0, 5 minus A.
- That's just 1, 2, 4, 3.
- So that is equal to the null space of the matrix.
- 5 minus 1 is 4.
- 0 minus 2 is minus 2.
- 0 minus 4 is minus 4.
- And then, 5 minus 3 is 2.
- So the null space of this matrix right here-- and this
- matrix is just an actual numerical representation of
- this matrix right here.
- The null space of this matrix is the set of all of the
- vectors that satisfy this or all of the eigenvectors that
- correspond to this eigenvalue.
- Or, the eigenspace that corresponds to
- the eigenvalue 5.
- These are all equivalent statements.
- So we just need to figure out the null space of this guy is
- all of the vectors that satisfy the equation 4 minus
- 2, minus 4, 2 times some eigenvector is
- equal to the 0 vector.
- And the null space of a matrix is equal to the null space of
- the reduced row echelon form of a matrix.
- So what's the reduced row echelon form of this guy?
- Well, I guess a good starting point-- let me keep my first
- row the same, 4 minus 2.
- And let me replace my second row with my second row
- plus my first row.
- So minus 4 plus 4 is 0.
- 2 plus minus 2 is 0.
- Now, let me divide my first row by 4 and I
- get 1, minus 1/2.
- And then I get 0, 0.
- So what's the null space of this?
- This corresponds to v.
- This times v1, v2-- that's just another way of writing my
- eigenvector v-- has got to be equal to the 0 vector.
- Or another way to say it is that my first entry v1, which
- corresponds to this pivot column, plus or minus 1/2
- times my second entry has got to be equal to
- that 0 right there.
- Or, v1 is equal to 1/2 v2.
- And so if I wanted to write all of the eigenvectors that
- satisfy this, I could write it this way.
- My eigenspace that corresponds to lambda equals 5.
- That corresponds to the eigenvalue 5 is equal to the
- set of all of the vectors, v1, v2, that are equal to some
- scaling factor.
- Let's say it's equal to t times what?
- If we say that v2 is equal to t, so v2 is going to be equal
- to t times 1.
- And then, v1 is going to be equal to 1/2 times v2
- or 1/2 times t.
- Just like that.
- For any t is a member of the real numbers.
- If we wanted to, we could scale this up.
- We could say any real number times 1, 2.
- That would also be the span.
- Let me do that actually.
- It'll make it a little bit cleaner.
- Actually, I don't have to do that.
- So we could write that the eigenspace for the eigenvalue
- 5 is equal to the span of the vector 1/2 and 1.
- So it's a line in R2.
- Those are all of the eigenvectors that satisfy--
- that work for the equation where the
- eigenvalue is equal to 5.
- Now what about when the eigenvalue is
- equal to minus 1?
- So let's do that case.
- When lambda is equal to minus 1, then we have-- it's going
- to be the null space.
- So the eigenspace for lambda is equal to minus 1 is going
- to be the null space of lambda times our identity matrix,
- which is going to be minus 1 and 0, 0, minus 1.
- It's going to be minus 1 times 1, 0, 0, 1, which is just
- minus 1 there.
- Minus A.
- So minus 1, 2, 4, 3.
- And this is equal to the null space of-- minus 1,
- minus 1 is minus 2.
- 0 minus 2 is minus 2.
- 0 minus 4 is minus 4 and minus 1 minus 3 is minus 4.
- And that's going to be equal to the null space of the
- reduced row echelon form of that guy.
- So we can perform some row operations right here.
- Let me just put it in reduced row echelon form.
- So if I replace my second row plus 2 times my first row.
- So I'll keep the first row the same.
- Minus 2, minus 2.
- And then my second row, I'll replace it with two times--
- I'll replace it with it plus 2 times the first. Or even
- better, I'm going to replace it with it plus minus 2 times
- the first. So minus 4 plus 4 is 0.
- And then if I divide the top row by minus 2, the reduced
- row echelon form of this matrix right here or this
- matrix right here is going to be 1, 1, 0.
- So the eigenspace that corresponds to the eigenvalue
- minus 1 is equal to the null space of this guy right here
- It's the set of vectors that satisfy this
- equation: 1, 1, 0, 0.
- And then you have v1, v2 is equal to 0.
- Or you get v1 plus-- these aren't vectors,
- these are just values.
- v1 plus v2 is equal to 0.
- Because 0 is just equal to that thing right there.
- So 1 times v1 plus 1 times v2 is going to be equal to that 0
- right there.
- Or I could write v1 is equal to minus v2.
- Or if we say that v2 is equal to t, we could say v1 is equal
- to minus t.
- Or we could say that the eigenspace for the eigenvalue
- minus 1 is equal to all of the vectors, v1, v2 that are equal
- to some scalar t times v1 is minus t and v2 is plus t.
- Or you could say this is equal to the span of the vector
- minus 1 and 1.
- So let's just graph this a little bit just to understand
- what we just did.
- We were able to find two eigenvalues for
- this, 5 and minus 1.
- And we were able to find all of the vectors that are
- essentially-- or, we were able to find the set of vectors
- that are the eigenvectors that correspond to each of these
- eigenvalues.
- So let's graph them.
- So if we go to R2, let me draw my axes, this
- is my vertical axis.
- That's my horizontal axis.
- So all of the vectors that correspond to lambda equal 5
- are along the line 1/2, 1.
- Or the span of 1/2, 1.
- So that is 1.
- That is 1.
- So you go 1/2 and 1 just like that.
- So that's that vector, spanning vector.
- But anything along the span of this, all the multiples of
- this, are going to be valid eigenvectors.
- So anything along that line, all of the vectors when you
- draw them in standard position, point to a
- point on that line.
- All of these vectors, any vector on there is going to be
- a valid eigenvector and the corresponding eigenvalue is
- going to be equal to 5.
- So you give me this guy right here.
- When you apply the transformation, it's going to
- be five times this guy.
- If this guy is x, t of x is going to be
- five times this guy.
- Whatever vector you give along this line, the transformation
- of that guy, the transformation is literally,
- multiplying it by the matrix A.
- Where did I have the matrix A?
- The matrix A right up there.
- You're essentially just scaling this guy by 5 in
- either direction.
- This is for lambda equal 5.
- And for lambda equals 1, it's the span of this vector, which
- is minus 1, 1.
- Which looks like this.
- So this vector looks like that.
- We care about the span of it.
- Any vector that when you draw in standard position lies, or
- points to, points on this line, will be an eigenvector
- for the eigenvalue minus 1.
- So lambda equals minus 1.
- Let's say you take the spanning vector here.
- You apply the transformation, you're going to get
- minus 1 times it.
- So if this is x, the transformation of x is going
- to be that right there.
- Same length, just in the opposite direction.
- If you have this guy right here, you apply the
- transformation, it's going to be in the same spanning line
- just like that.
- So the two eigenspaces for the matrix-- where did I write it?
- I think it was the matrix 1, 2, 3-- 1, 2, 4, 3.
- The two eigenvalues were 5 and minus 1.
- And then it has an infinite number of eigenvectors, so
- they actually create two eigenspaces.
- Each of them correspond to one of the eigenvalues.
- And these lines represent those two eigenspaces.
- You give me any vector in either of these sets and
- they're going to be an eigenvector.
- I'm using the word vector too much.
- You give me any vector in either of these sets, and they
- will be an eigenvector for our matrix A.
- And then, depending on which line it is, we know what their
- transformation is going to be.
- If it's going to be on this guy, we take the
- transformation, the resulting vector's going to be five
- times the vector.
- If you take one of these eigenvectors and you transform
- it, the resulting transformation of the vector's
- going to be minus 1 times that vector.
- Anyway, we now know what eigenvalues, eigenvectors,
- eigenspaces are.
- And even better, we know how to actually find them.
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At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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