Now I want to talk about the H matrix.
This is a matrix that takes a state, and when it multiplies
by that state, spits out a measurement.
Remember, we can only directly measure position and velocity,
so that's all we want the H matrix to keep.
Again, I want to talk about the 2D lecture case and the 4D homework case.
Hopefully, by comparing them, we'll be able to build some intuition,
and you'll be able to answer the homework.
What was the goal of the H matrix?
The goal of the H matrix was to take some state--
in the 2D case, our state was represented as an x and an ẋ--
multiply some matrix by that state in such a way that we extract a measurement.
In the 2D case the measurement was just x--just the x coordinate.
We can think of this as a 1 x 1 vector or a 1 x 1 matrix.
The matrix we use to do that was this one.
That was our H matrix--1, 0--because 1 times x gives us the x,
and 0 times ẋ gives us the nothing--exactly what we want.
But now let's talk about the dimensionality of these matrices
and how this multiplication yielded just this number x.
So we can think of x here as a 1 x 1 matrix.
We got that matrix by multiplying this one, which is a 1 x 2--
one row by two columns--with this, which is two rows by one column.
What we see here is that this 1 actually came from right here,
and this 1 came from right here.
These 2s we can think of as canceling out, in a way,
giving us this 1 x 1 matrix.
Now, let's see if we can generalize that to the 4-dimensional case as presented in the homework.
In the 4-dimensional case our state is now given by x, y, ẋ, ẏ.
We're going to have some H matrix.
I don't know anything about it yet, but I'm just going to put this there for now as a placeholder.
We want to get a measurement from that. What should this measurement be?
It's not just going to be x, because now our position includes both x and y.
So it's going to be a column vector--x and y.
Again, let's think. What's going on with the dimensionality here?
Here we have a 2 x 1 matrix,
and that came from this matrix, which I said we don't know anything about yet--
I'll just say a question mark by question mark--
and this matrix, which is four rows by one column.
Now, can you use the intuition we built up here
for how the dimensionality of matrices works with this to fill in the question marks?
Once you figure out the number of rows and the number of columns in this H matrix,
figuring out where to put your 1s and 0s will be a little bit easier.
I wish you luck.