0:00:00.000,0:00:02.000
Now I want to talk about the H matrix.
0:00:02.000,0:00:05.000
This is a matrix that takes a state, and when it multiplies
0:00:05.000,0:00:08.000
by that state, spits out a measurement.
0:00:08.000,0:00:11.000
Remember, we can only directly measure position and velocity,
0:00:11.000,0:00:14.000
so that's all we want the H matrix to keep.
0:00:14.000,0:00:19.000
Again, I want to talk about the 2D lecture case and the 4D homework case.
0:00:19.000,0:00:21.000
Hopefully, by comparing them, we'll be able to build some intuition,
0:00:21.000,0:00:24.000
and you'll be able to answer the homework.
0:00:24.000,0:00:26.000
What was the goal of the H matrix?
0:00:26.000,0:00:31.000
The goal of the H matrix was to take some state--
0:00:31.000,0:00:35.000
in the 2D case, our state was represented as an x and an ẋ--
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multiply some matrix by that state in such a way that we extract a measurement.
0:00:41.000,0:00:48.000
In the 2D case the measurement was just x--just the x coordinate.
0:00:48.000,0:00:52.000
We can think of this as a 1 x 1 vector or a 1 x 1 matrix.
0:00:52.000,0:00:56.000
The matrix we use to do that was this one.
0:00:56.000,0:01:01.000
That was our H matrix--1, 0--because 1 times x gives us the x,
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and 0 times ẋ gives us the nothing--exactly what we want.
0:01:05.000,0:01:08.000
But now let's talk about the dimensionality of these matrices
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and how this multiplication yielded just this number x.
0:01:13.000,0:01:20.000
So we can think of x here as a 1 x 1 matrix.
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We got that matrix by multiplying this one, which is a 1 x 2--
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one row by two columns--with this, which is two rows by one column.
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What we see here is that this 1 actually came from right here,
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and this 1 came from right here.
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These 2s we can think of as canceling out, in a way,
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giving us this 1 x 1 matrix.
0:01:46.000,0:01:50.000
Now, let's see if we can generalize that to the 4-dimensional case as presented in the homework.
0:01:50.000,0:01:58.000
In the 4-dimensional case our state is now given by x, y, ẋ, ẏ.
0:01:58.000,0:02:00.000
We're going to have some H matrix.
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I don't know anything about it yet, but I'm just going to put this there for now as a placeholder.
0:02:05.000,0:02:09.000
We want to get a measurement from that. What should this measurement be?
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It's not just going to be x, because now our position includes both x and y.
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So it's going to be a column vector--x and y.
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Again, let's think. What's going on with the dimensionality here?
0:02:23.000,0:02:26.000
Here we have a 2 x 1 matrix,
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and that came from this matrix, which I said we don't know anything about yet--
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I'll just say a question mark by question mark--
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and this matrix, which is four rows by one column.
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Now, can you use the intuition we built up here
0:02:44.000,0:02:51.000
for how the dimensionality of matrices works with this to fill in the question marks?
0:02:51.000,0:02:56.000
Once you figure out the number of rows and the number of columns in this H matrix,
0:02:56.000,0:02:59.000
figuring out where to put your 1s and 0s will be a little bit easier.
0:02:59.000,9:59:59.000
I wish you luck.