Orthogonal matrices preserve angles and lengths Showing that orthogonal matrices preserve angles and lengths
Orthogonal matrices preserve angles and lengths
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- In the last couple of videos, we've seen that if we have
- some matrix C that is n by n.
- It's a square matrix, and is columns, column form and
- orthonormal set.
- Which just means that the columns each have been
- So they each have length of 1 if you view
- them as column vectors.
- And they're all mutually orthogonal to each other.
- So if you dot it with yourself you get 1.
- If you dot it with any of the other columns, you get 0.
- We've seen this multiple times.
- It's orthogonal to everything else.
- If you have a matrix like this-- and I actually forgot
- to tell you the name of this-- this is called
- an orthogonal matrix.
- We've already seen that the transpose of this matrix is
- the same thing as the inverse of this matrix.
- Which makes it super, duper, duper useful to deal with.
- The transpose of this matrix is equal to the inverse.
- Now, this statement leads to some other interesting things
- about this.
- So, so far we've been dealing this mainly with
- the change of basis.
- I can kind of draw the diagram that you're
- probably tired of by now.
- Let's say that's the standard basis.
- Let's say that I have x in coordinates
- with another basis.
- We've seen I can multiply this guy times c.
- To get that up there I could multiply that guy by c inverse
- to get this guy right here.
- And, in that world, we viewed c as just a change of basis.
- Were representing the same matrix-- we're representing
- the same vector.
- We're just changing the coordinates of how we
- represent it.
- But we also know that any matrix product, any matrix
- vector product, is also a linear transformation.
- So, this change of basis is really just a linear
- What I want to show you in this video, and you could view
- it either as a change of basis or as a linear transformation,
- is that when you multiply this orthogonal matrix times some
- vector, it preserves-- let me write this
- down-- lengths and angles.
- So let's have a little touchy-feely discussion of
- what that means.
- Let's view it as a transformation.
- Let's say I have some set of vectors in my domain.
- Let's say they look like this.
- Let's say that it looks like this.
- Well, let me do it like-- I'll draw that one like that guy,
- and this guy like that.
- And there's some angle between them.
- Angles are easy to visualize in r2, r3.
- Maybe a little harder once we get the higher dimensions.
- But that's the angle between them.
- Now, if we're saying that we're preserving the angles
- and the lengths, that means if I were to multiply these
- vectors times c then we could view it as a transformation.
- Maybe I rotate them or I-- well, you can't really.
- Maybe I rotate them or do something like that.
- So maybe that pink vector will now look like this.
- But it's going to have the same length.
- This length is going to be the same thing as that length.
- And even more, when I said it preserves lengths and angles,
- this yellow vector's going to look something like this.
- Where the angle is going to be the same.
- Where this data is going to be that data.
- That's what I mean by preserves angles.
- If we didn't have this case, we could imagine a
- transformation that doesn't preserve angles.
- Let me draw one that doesn't.
- If this got transformed to, I don't know, let's say this guy
- got a lot longer, and let's say this guy also got longer,
- and I want to show that the angle
- also doesn't get preserved.
- Not only did it get longer, but it got
- distorted a little bit.
- So, the angle also changed.
- This transformation right there is
- not preserving angles.
- So when you have a change of basis matrix that's
- orthogonal, when you have a transformation matrix that's
- orthogonal, all it's essentially doing to your to
- your vectors, is it kind of a rotates them around, but it's
- not going to really distort them.
- So I'll write that in quotes because that's not a
- mathematically rigorous term.
- So, no distortion of vectors.
- So, I've kind of showed you the intuition
- of what that means.
- Let's actually prove it to ourselves
- that this is the case.
- So, I'm saying that if this pink vector here is x, and
- that this pink vector here is c times x, I'm claiming that
- the length of x is equal to the length of c times x.
- Let's see if that's actually the case.
- The length of cx squared is the same thing as cx dot cx.
- And here it's always useful for me to kind of remind
- myself that if I take two vectors-- let
- me do it over here.
- Let's say I have y dot y.
- This is the same thing as y transpose, if you view them as
- matrices, y transpose times y.
- y transpose y is just y1, y2, all the way to yn times y1,
- y2, all the way to yn.
- And if you were to do this 1 by n times n by 1 matrix
- product, you're going to get a 1 by 1 matrix or just a number
- that's going to be y1 times y1 plus y2 times y2 all the way
- to yn times yn.
- So, this is the same thing as y dot y.
- I think I did this about ten or twenty videos ago, but it's
- always a good refresher.
- So let's use this property right here.
- So these two dotted with each other.
- This is the same thing is taking one of their transpose
- times the other one.
- So turn this from a vector, vector dot product to a
- matrix, matrix product.
- So this is the same thing as CX transpose, CX.
- so you can view this as a 1 by n matrix now, times the 1 by 1
- matrix which is just the column vector cx.
- These are the same thing.
- Now, we also know that A times B transpose is the same thing
- is B transpose, A transpose.
- We saw that a long time ago.
- So this thing right here is going to be equal to X
- transpose, C transpose.
- Just switch the order and take the transpose of each.
- X transpose times C transpose.
- And then you have that times CX.
- And now we know that C transpose is the same thing is
- as C inverse.
- This is where we need the orthogonality of the matrix C.
- This is where we need it to be a square matrix where all of
- its columns are mutually orthogonal
- and they're all normal.
- And so this thing is just going to become
- the identity matrix.
- I can write the identity matrix there, but that's just
- going to disappear.
- So this is going to be equal to X transpose X.
- X transpose is the same thing as X dot X which is the same
- thing as the length of X squared.
- So the length of CX squared is the same thing as the length
- of X squared.
- So, that tells us that the length of X, or the length of
- CX, is the length of x because both of these are going to be
- positive quantities.
- So I've shown you that orthogonal matrices definitely
- preserve length.
- Let's see if they preserve angles.
- So we actually have to define angles.
- Throughout our mathematical careers, we understood what
- angles mean in kind of r2 or r3.
- But in linear algebra, we like to be general.
- And we defined an angle using the dot product.
- We use the law of cosines and we took an analogy to kind of
- triangle in r2.
- But we defined an angle or we said the dot product V dot W
- is equal to the lengths, the products of the lengths of
- those two vectors times the cosine of the
- angle between them.
- Or you could say that the cosine of the angle between
- two vectors, we defined as the dot product of those two
- vectors divided by the lengths of those two vectors.
- This was the definition so that we can extend the idea of
- an angle to an arbitrarily high dimension to r
- google if we had to.
- So let's see if it preserves.
- Let's see what the angle is if we multiply these guys by C.
- So, if we wanted-- let's say our new angle.
- So, cosine of angle C.
- Once we perform our transformation.
- We're going to perform the transformation on all of these
- It's going to be CV dot CW over the lengths of CV times
- the lengths of CW.
- Now we already know that lengths are preserved.
- We already know that the length of CW and CV are just
- going to be W and V.
- We just proved that.
- Let me write that.
- So the cosine of theta C is equal to CV dot CW over the
- lengths of V times W.
- Because we've already shown that it preserves lengths.
- We'll see what this top part equals.
- So we can just use the general property.
- The dot product is equal to the transpose of one guy as
- kind of a matrix times the second guy.
- So this is equal to CW transpose times CV.
- And all of that over these lengths.
- Like the W.
- And this is going to be equal to-- I'm going to write it
- down here to have some space.
- We can switch these guys and take their transpose.
- So, it's W transpose times C transpose times CV.
- All of that over their lengths, the product of their
- lengths V and W.
- And this is the identity matrix.
- That's the identity matrix, and this is going to be equal
- to W transpose times V over the products of their lengths.
- And this is the same thing as V dot W.
- This is V dot W over their lengths.
- Which is cosine of theta.
- So, you notice, by our definition of an angle as the
- dot product divided by the vector lengths, when you
- perform a transformation or you can imagine a change of
- basis either way, with an orthogonal matrix C the angle
- between the transformed vectors does not change.
- It is the same as the angle between the vectors before
- they were transformed.
- Which is a really neat thing to know.
- The change of bases or transformations with
- orthogonal matrices don't distort the vectors.
- They might just kind of rotate them around or shift them a
- little bit, but it doesn't change the
- angles between them.
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