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
Showing that an eigenbasis makes for good coordinate systems Showing that an eigenbasis makes for good coordinate systems
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- I've talked a lot about the idea that eigenvectors could
- make for good bases or good basis vectors.
- So let's explore that idea a little bit more.
- Let's say I have some transformation.
- Let's say it's a transformation from Rn to Rn,
- and it can be represented by the matrix, A.
- So the transformation of x is equal to the n-by-n
- matrix, A times x.
- Now let's say that we have n linearly independent
- eigenvectors of A.
- And this isn't always going to be the case, but it can often
- be the case.
- It's definitely possible.
- Let's assume that A has n linearly independent
- eigenvectors.
- So I'm going to call them v1, v2, all the way through vn.
- Now, n linearly independent vectors in Rn can definitely
- be a basis for Rn.
- We've seen that multiple times.
- And what I want to show you in this video is that this makes
- a particularly good basis for this transformation.
- So let's explore that.
- So the transformation of each of these vectors-- I'll write
- it over here.
- The transformation of vector 1 is equal to A times vector 1
- and since vector 1 is an eigenvector of A, that's going
- to be equal to some eigenvalue lambda 1 times vector 1.
- We could do that for all of them.
- The transformation of vector 2 is equal to A times v2, which
- is equal to some eigenvalue lambda 2 times v2.
- And I'm just going to skip all of them and just go straight
- to the nth one.
- We have n of these eigenvectors.
- You might have a lot more.
- We're just assuming that A has at least n linearly
- independent eigenvectors.
- In general, you could take scaled up versions of these
- and they'll also be eigenvectors.
- Let's see, so the transformation of vn is going
- to be equal to A times vn.
- And because these are all eigenvectors, A times vn is
- just going to be lambda n, some eigenvalue times the
- vector, vn.
- Now, what are these also equal to?
- Well, this is equal to, and this is probably going to be
- unbelievably obvious to you, but this is the same thing as
- lambda 1 times vn plus 0 times v2 plus all the
- way to 0 times vn.
- And this right here is going to be 0 times v1 plus lambda 2
- times v2 plus all the way, 0 times all of the
- other vectors vn.
- And then this guy down here, this is going to be 0 times v1
- plus 0 times v2 plus 0 times all of these basis vectors,
- these eigenvectors, but lambda n times vn.
- This is almost stunningly obvious, right?
- I just rewrote this as this plus a bunch of zero vectors.
- But the reason why I wrote that is, because in a second,
- we're going to take this as a basis and we're going to find
- coordinates with respect to that basis, and so this guy's
- coordinates will be lambda 1, 0, 0, because that's the
- coefficients on our basis vectors.
- So let's do that.
- So let's say that we define this as some basis.
- So B is equal to the set of-- actually, I don't even have to
- write it that way.
- Let's say I say that B, I have some basis B,
- that's equal to that.
- What I want to show you is that when I do a change of
- basis-- we've seen this before-- in my standard
- coordinates or in coordinates with respect to the standard
- basis, you give me some vector in Rn, I'm going to multiply
- it times A, and you're going to have the
- transformation of it.
- It's also going to be in Rn.
- Now, we know we can do a change of basis.
- And in a change of basis, if you want to go that way, you
- multiply by C inverse, which is-- remember, the change of
- basis matrix C, if you want to go in this direction, you
- multiply by C.
- The change of basis matrix is just a matrix with all of
- these vectors as columns.
- It's very easy to construct.
- But if you change your basis from x to our new basis, you
- multiply it by the inverse of that.
- We've seen that multiple times.
- If they're all orthonormal, then this is the same thing as
- a transpose.
- We can't assume that, though.
- And so this is going to be x in our new basis.
- And if we want to find some transformation, if we want to
- find the transformation matrix for T with respect to our new
- basis, it's going to be some matrix D.
- And if you multiply D times x, you're going to get this guy,
- but you're going to get the B representation of that guy.
- The transformation of the vector x is B representation.
- And if we want to go back and forth between that guy and
- that guy, if we want to go in this direction, you can
- multiply this times C, and you'll just get the
- transformation of x.
- And if you want to go in that direction, you could multiply
- by the inverse of your change of basis matrix.
- We've seen this multiple times already.
- But what I've claimed or I've kind of hinted at is that if I
- have a basis that's defined by eigenvectors of A, that this
- will be a very nice matrix, that this might be the
- coordinate system that you want to operate in, especially
- if you're going to apply this matrix a lot.
- If you're going to do this transformation on a lot of
- different things, you're going to do it over and over and
- over again, maybe to the same set, then it maybe is worth
- the overhead to do the conversion and just use this
- as your coordinate system.
- So let's see that this will actually be a nice-looking,
- easy-to-compute-with and actually diagonal matrix.
- So we know that the transformation-- what is the
- transformation of-- let's write this in a bunch of
- different formats.
- Let me scroll down a little bit.
- So if I wanted to write the transformation of v1 in B
- coordinates, what would it be?
- It's just going to be equal to-- well, these are the basis
- vectors, right?
- So it's the coefficient on the basis vectors.
- So it's going to be equal to lambda 1, and then there's a
- bunch of zeroes.
- It's lambda 1 times v1 plus 0 times v2 plus 0 times v3, all
- the way to 0 times vn.
- That's what it's equal to.
- But it's also equal to D, and we can write D like this.
- D is also a transformation between Rn and Rn, just a
- different coordinate system.
- So D is going to just be a bunch of column vectors d1,
- d2, all the way through dn times-- this is the same thing
- as D times our B representation
- of the vector v1.
- But what is our B representation
- of the vector v1?
- Well, the vector, v1 is just 1 times v1 plus 0 times v2 plus
- 0 times v3 all the way to 0 times vn.
- v1 is a basis vector.
- That's just 1 times itself plus 0 times everything else.
- So this is what its representation is in the B
- coordinate system.
- Now, what is this going to be equal to?
- And we've seen this before.
- This is all a bit of review.
- I might even be boring you.
- This is just equal to 1 times d1 plus 0 times d2 plus 0
- times all the other columns.
- This is just equal to d1.
- So just like that, we have our first column of our matrix D.
- We could just keep doing that.
- I'll do it multiple times.
- The transformation of v2 in our new coordinate system with
- respect to our new basis is going to be equal to-- well,
- we know what the transformation of v2 is.
- It's 0 times v1 plus lambda 2 times v2 and then plus 0 times
- everything else.
- And that's the same thing as D, d1, d2, all the way through
- dn times our B representation of vector 2.
- Well, vector 2 is one of the basis vectors.
- It's just 0 times v1 plus 1 times v2 plus 0 times v3 all
- the way, the rest is 0.
- So what's this going to be equal to?
- This is 0 times d1 plus 1 times d2 and 0 times
- everything else, so it's equal to d2.
- I think you get the general idea.
- I'll do it one more time just to really
- hammer the point home.
- The transformation of the nth basis vector, which is also an
- eigenvector of our original matrix A or of our
- transformation in standard coordinates, in B coordinates,
- is going to be equal to what?
- Well, we wrote it right up here.
- It's going to be a bunch of zeroes.
- It's 0 times all of these guys plus lambda n times vn.
- And this is going to be this guy d1, d2, all the way to dn
- times the B representation of the nth basis vector, which is
- just 0 times v1, 0 times v2 and 0 times all of them,
- except for 1 times vn.
- And so this is going to be equal to 0 times d1 plus 0
- times d2 plus 0 times all of these guys all the
- way to 1 times dn.
- So that's going to be equal to dn.
- And just like that, we know what our transformation matrix
- is going to look like with respect to this new basis,
- where this basis was defined or it's made up of n linearly
- independent eigenvectors of our original matrix A.
- So what does D look like?
- Our matrix D is going to look like-- its first column is
- right there.
- We figured that one out.
- Lambda 1, and then we just have a bunch of zeroes.
- Its second column is right here.
- d2 is equal to this.
- It's 0, lambda 2, and then a bunch of zeroes.
- And then this is in general the case.
- The nth column is going to have a zero everywhere except
- along the diagonal.
- It's going to be lambda n.
- It's going to be the eigenvalue for the nth
- eigenvector.
- And so the diagonal is going to look-- you're going to have
- lambda 3 all the way down to lambda n.
- And our nth column is lambda n with just a bunch of zeroes
- everywhere.
- So D, when we picked-- this is a neat result.
- If A has n linearly independent eigenvectors, and
- this isn't always the case, but we can figure out that
- eigenvectors and say, hey, I can take a collection of n of
- these that are linearly independent, then those will
- be a basis for Rn.
- n linearly independent vectors in Rn are a basis for Rn.
- But when you use that basis, when you use the linearly
- independent eigenvectors of A as a basis, we call this an
- eigenbasis.
- The transformation matrix with respect to that eigenbasis, it
- becomes a very, very nice matrix.
- This is super easy to multiply.
- It's super easy to invert.
- It's super easy to take the determinant of.
- We've seen it multiple times.
- It just has a ton of neat properties.
- It's just a good basis to be dealing with.
- So that's kind of the big takeaway.
- In all of linear algebra, we did all this stuff with spaces
- and vectors and all of that, but in general, vectors are
- abstract representations of real world things.
- You could represent a vector as the stock returns or it
- could be a vector of weather in a certain part of the
- country, and you can create these spaces based on the
- number of dimensions and all of that.
- And then you're going to have transformations.
- Sometimes, like when we learn about Markov chains, your
- transformations are essentially what's the
- probability after one time increment that something state
- will change to something else, then you'll want to apply that
- matrix many, many, many, many times to see what the stable
- state is for a lot of things.
- And I know I'm not explaining any of this to you well, but I
- wanted to tell you that all of linear algebra is really just
- a very general way to solve a whole universe of problems.
- And what's useful about this is you can have transformation
- matrices that define these functions
- essentially on data sets.
- And what we've learned now is that when you look at the
- eigenvectors and the eigenvalues, you can change
- your bases so that you can solve your problems in much
- simpler ways.
- And I know it's all very abstract right now, but you
- now have the toolkit, and the rest of your life, you have to
- figure out how to apply this toolkit to specific problems
- in probability or statistics or finance or modeling weather
- systems or who knows what else.
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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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