Introduction to Eigenvalues and Eigenvectors What eigenvectors and eigenvalues are and why they are interesting
Introduction to Eigenvalues and Eigenvectors
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- For any transformation that maps from Rn to Rn, we've done
- it implicitly, but it's been interesting for us to find the
- vectors that essentially just get scaled up by the
- So the vectors that have the form-- the transformation of
- my vector, is just equal to some scaled-up
- version of a vector.
- And if this doesn't look familiar, I can jog your memory
- a little bit.
- When we were looking for basis vectors for the
- transformation-- let me draw it.
- This was from R2 to R2,
- from R2 to R2.
- So let me draw R2 right here.
- Now let's say I had the vector...let's add the vector...let's say v1 was equal to
- the vector 1, 2.
- And we had the lines spanned by that vector.
- We did this problem several videos ago.
- And I had the transformation that flipped across this line.
- So if we call that line l, T was the transformation from R2
- to R2 that flipped vectors across this line.
- So it flipped, flipped vectors, flipped vectors, across l.
- So if you remember that transformation, if I had some
- random vector that looked like that, let's say that's x,
- that's vector x, then the transformation of x looks
- something like this.
- Which is flipped across that line.
- That was the transformation of x.
- And, if you remember that video, we were looking for a
- change of basis that would allow us to at least figure
- out the, the matrix for the transformation, at least an
- alternate basis.
- And then we could figure out the matrix for the
- transformation in the standard basis.
- And the basis we picked were basis vectors that didn't get
- changed much by the transformation, or ones that
- only got scaled by the transformation.
- For example, when I took the transformation of v1, when I took the transformation of v1, it just
- equaled to v1.
- Or we could say, that the transformation of v1, just
- equaled 1 times v1.
- So if you just follow this, this little format that I set up
- here, lambda, in this case, would be 1.
- And of course, the vector in this case is v1.
- The transformation just scaled up v1 by 1.
- Now if you also or if you, that same problem, we had the other vector that
- we also looked at.
- OK, it was the vector... it was the vector...minus let's say it's the vector v2,
- which is-- let's say it's 2 minus 1.
- And then if you take the transformation of it, since it
- was orthogonal to the line, it just got
- flipped over like that.
- And that was a pretty interesting vector force as
- well, because the transformation of v2 in this
- situation is equal to what?
- Just minus v2.
- It's equal to minus v2.
- Or, you could say, that the transformation of v2 is equal
- to minus 1 times v2.
- And these were interesting vectors for us because when we
- defined a new basis with these guys as the basis vector, it
- was very easy to figure out our transformation matrix.
- And actually, that basis was very easy to compute with.
- And we'll explore that a little bit more in the future.
- But hopefully you realize that these are interesting vectors.
- There was also the cases where we had the planes spanned by
- some vectors.
- And then we had another vector that was popping out of the
- plane like that.
- And we were transforming things by taking the mirror
- image across this and we're like, well in that
- transformation, these red vectors don't change at all
- and this guy gets flipped over.
- So maybe those would make for good bases.
- Or those would make for good basis vectors.
- And they did.
- So in general, we're always interested with the vectors
- that just get scaled up by a transformation.
- It's not going to be all vectors, right?
- This vector that I drew here, this vector x, it doesn't just
- get scaled up, it actually gets changed, this direction
- gets changed.
- The vectors that get scaled up might switch direct-- might go
- from this direction to that direction, or maybe
- they go from that.
- Maybe that's x and then the transformation of x might be a
- scaled up version of x.
- Maybe it's that.
- The actual, I guess, line that they span will not change.
- And so that's what we're going to concern ourselves with.
- These have a special name.
- And they have a special name and I want to make this very
- clear because they're useful.
- It's not just some mathematical game we're
- playing, although sometimes we do fall into that trap.
- But they're actually useful.
- They're useful for defining bases because in those bases
- it's easier to find transformation matrices.
- They're more natural coordinate systems. And
- oftentimes, the transformation matrices in those bases are
- easier to compute with.
- And so these have special names.
- Any vector that satisfies this right here is called an
- eigenvector for the transformation T.
- And the lambda, the multiple that it becomes-- this is the
- eigenvalue associated with that eigenvector.
- So in the example I just gave where the transformation is
- flipping around this line, v1, the vector 1, 2 is an
- eigenvector of our transformation.
- So 1, 2 is an eigenvector.
- And it's corresponding eigenvalue is 1.
- This guy is also an eigenvector-- the
- vector 2, minus 1.
- He's also an eigenvector.
- A very fancy word, but all it means is a vector that's just
- scaled up by a transformation.
- It doesn't get changed in any more meaningful way than just
- the scaling factor.
- And it's corresponding eigenvalue is minus 1.
- If this transformation-- I don't know what its
- transformation matrix is.
- I forgot what it was.
- We actually figured it out a while ago.
- If this transformation matrix can be represented as a matrix
- vector product-- and it should be; it's a linear
- transformation-- then any v that satisfies the
- transformation of-- I'll say transformation of v is equal
- to lambda v, which also would be-- you know, the
- transformation of [? v ?]
- would just be A times v.
- These are also called eigenvectors of A, because A
- is just really the matrix representation of the
- So in this case, this would be an eigenvector of A, and this
- would be the eigenvalue associated with the
- So if you give me a matrix that represents some linear
- You can also figure these things out.
- Now the next video we're actually going to figure out a
- way to figure these things out.
- But what I want you to appreciate in this video is
- that it's easy to say, oh, the vectors that
- don't get changed much.
- But I want you to understand what that means.
- It literally just gets scaled up or maybe they get reversed.
- Their direction or the lines they span
- fundamentally don't change.
- And the reason why they're interesting for us is, well,
- one of the reasons why they're interesting for us is that
- they make for interesting basis vectors-- basis vectors
- whose transformation matrices are maybe computationally more
- simpler, or ones that make for better coordinate systems.
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