Where did this step come from (linear algebra/vector calculus)

[astral flax]

This is talking about the fact that you can take z = xy, and use matrices and quadratic forms to "rotate" the coordinate system by an angle, theta, such that z = xy takes the form of another function that makes sense for later steps. But where in the world did the "Well, if we rotate coordinates:" portion come from? That went from 0 to 100 so fast for me. Can someone explain the structure of the expression?

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[meager crater]

$\begin{pmatrix}
cos(\theta) & -sin(\theta) \
sin(\theta) & cos(\theta) \
\end{pmatrix}$ is known as the 2d rotation matrix.

[haughty igloo]

Yojda

[meager crater]

That means when you compute this matrix, you'll get your basis rotated.

[meager crater]

with $-sin(\theta) = cos(\theta + \pi/2)$ and $cos(\theta) = sin(\theta + \pi/2)$

[haughty igloo]

Yojda

[astral flax]

Yeah, I understand how that matrix causes a rotation, but I don't understand how they used that fact to convert the matrix form of z = xy to a rotated coordinate form

The substitution makes no sense to me

Why are the two rotation matrices different here?

[meager crater]

Cause you have $\begin{pmatrix}
x & y
\end{pmatrix}$ and $\begin{pmatrix}
x \
y
\end{pmatrix}$ vectors

[haughty igloo]

Yojda

[astral flax]

Okay, that's a step closer to what's confusing me. What's the difference?

How do the multiplications work out differently?

I'm only used to a matrix being multiplied by a column vector, I don't know what to do with a row vector

[meager crater]

Because there are some multiplication rules about the size of your matrices

You can't multiply $\begin{pmatrix}
x \
y
\end{pmatrix} \begin{pmatrix}
cos(\theta) & -sin(\theta) \
sin(\theta) & cos(\theta)
\end{pmatrix}$

[haughty igloo]

Yojda

[meager crater]

So we write $\begin{pmatrix}
x & y
\end{pmatrix} \begin{pmatrix}
cos(\theta) & -sin(\theta) \
sin(\theta) & cos(\theta)
\end{pmatrix}$

[haughty igloo]

Yojda

[meager crater]

Now does $\begin{pmatrix}
x & y
\end{pmatrix}$ and $\begin{pmatrix}
x \
y
\end{pmatrix}$ represent the same vector? yes

[haughty igloo]

Yojda

[astral flax]

Making a little more sense. Why did they write (x' y') in the substitution? I thought x' and y' were the coordinates that were rotated or something?

Where I'm getting stuck is thinking that you can somehow show that the substitution equals the original expression with x and y

Am I misinterpreting something

[meager crater]

Okay, you have to think about what represents the matrix and what represents the vector

A matrix = your basis

A vector = just a vector

When you multiply a vector by a matrix (to the left), it means you represent your vector in another basis.

[astral flax]

So then x' and y' simply symbolize that they're altered by the rotation?

[meager crater]

Somehow

[astral flax]

Because the way I'm trying to make sense of the expression is like, how did $\begin{pmatrix}
x & y
\end{pmatrix}$ become $\begin{pmatrix}
x' & y'
\end{pmatrix}$ $\begin{pmatrix}
cos(\theta) & -sin(\theta)
sin(\theta) & cos(\theta)
\end{pmatrix}$

[haughty igloo]

Noe

[astral flax]

Sorry for the bad formatting

[meager crater]

Np

[astral flax]

And similarly for the column (x y) becoming the piece to the right

Because what I'm seeing is "before equals after", but...how?

[meager crater]

Because $\begin{pmatrix}
cos(\theta) & -sin(\theta) \
sin(\theta) & cos(\theta)
\end{pmatrix} \begin{pmatrix}
x \
y
\end{pmatrix} = \begin{pmatrix}
x' \
y'
\end{pmatrix}$

[haughty igloo]

Yojda

[astral flax]

Alright, now how does this work out with the weird row vector multiplication rule you mentioned before

Because if that can explain the correspondence, that may help

Also, wait a second. To get a valid substitution for the column (x y), wouldn't you need to multiply both sides by the inverse of that rotation matrix? To isolate the column (x y)

Wait a second...is this matrix the inverse of the rotation matrix?

[meager crater]

It's not a problem, the rotation matrix is orthogonal, so $Q^{-1} = Q^T$

[haughty igloo]

Yojda

[astral flax]

WHAT

That's amazing, why didn't I think of that

Of course it's the inverse

Oh my gosh, wait, so let me make sure I'm understanding this

Still a bit foggy. This basically is a direct substitution using the definition of those rotated coordinates. For the first one, the row (x y), it just has to be in that order because row count of the right matrix must equal the column count of the left matrix

And the column (x y) on the end becomes what it does with just an inverse

Because rotation times inverse rotation just takes you back

Okay yeah this is much smoother now

[meager crater]

Now you can detail those substitutions to be sure you understand (that's not difficult)

[haughty igloo]

Yojda

Yojda

Yojda

Noe

[astral flax]

Plugging in x' and y' will give me values of z all the same, right? It's just now it's taking into account the rotation of the coordinate system with the extra variable, theta

I think I finally understand where it came from a bit better

How does the entire expression become a value in the end, by the way? Is $\begin{pmatrix}
x & y
\end{pmatrix} \begin{pmatrix}
x \ y
\end{pmatrix}$ syntactically a dot product somehow?

How do I make it a column :/

[haughty igloo]

Noe

[astral flax]

There, double backslash

Had a feeling the escape was wrong. But yeah, is it?

[meager crater]

Yes it is, because you have x^2 + y^2.

[astral flax]

Can you "kinda" see it as matrix multiplication that results in a 1x1 matrix?

Like does that even make sense

[meager crater]

Sure

[astral flax]

Oh

[meager crater]

I thought this too.

That's maybe an abuse of notation

But in fact, a 1x1 matrix could be interpreted as a scalar, in the same way as a 1xn matrix could be interpreted as a vector.

That's what I think

Anyway, you can see this as the "definition of a dot product"

[astral flax]

I'm so not used to this abstract stuff

Things like this make me question what the definition of things even are anymore

[meager crater]

https://www.youtube.com/watch?v=fNk_zzaMoSs&list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab

So take a look at this

It saved me on my math courses

[astral flax]

I actually did watch this a while ago

Which is why I'm thinking back at the "just a collection of numbers" outlook he mentioned

[meager crater]

Yes, but you have all the series on linear algebra, not just this video

For example : https://www.youtube.com/watch?v=P2LTAUO1TdA&list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab&index=13

Multiply a vector by a matrix = change basis

[astral flax]

@meager crater For context, this was all inside an explanation for deriving the Hessian matrix from quadratic approximations in several variables

[meager crater]

Okay, I didn't do this before. That's why I don't understand the end, with the saddle. But sounds interesting.

Hope I helped you to understand this linear algebra part.

[astral flax]

You absolutely helped a lot, thanks so much

[keen flare]

@astral flax has given 1 rep to @meager crater

[astral flax]

+close