[Script Info]
Title:
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.00,0:00:02.00,Default,,0000,0000,0000,,Now I want to talk about the H matrix.
Dialogue: 0,0:00:02.00,0:00:05.00,Default,,0000,0000,0000,,This is a matrix that takes a state, and when it multiplies
Dialogue: 0,0:00:05.00,0:00:08.00,Default,,0000,0000,0000,,by that state, spits out a measurement.
Dialogue: 0,0:00:08.00,0:00:11.00,Default,,0000,0000,0000,,Remember, we can only directly measure position and velocity,
Dialogue: 0,0:00:11.00,0:00:14.00,Default,,0000,0000,0000,,so that's all we want the H matrix to keep.
Dialogue: 0,0:00:14.00,0:00:19.00,Default,,0000,0000,0000,,Again, I want to talk about the 2D lecture case and the 4D homework case.
Dialogue: 0,0:00:19.00,0:00:21.00,Default,,0000,0000,0000,,Hopefully, by comparing them, we'll be able to build some intuition,
Dialogue: 0,0:00:21.00,0:00:24.00,Default,,0000,0000,0000,,and you'll be able to answer the homework.
Dialogue: 0,0:00:24.00,0:00:26.00,Default,,0000,0000,0000,,What was the goal of the H matrix?
Dialogue: 0,0:00:26.00,0:00:31.00,Default,,0000,0000,0000,,The goal of the H matrix was to take some state--
Dialogue: 0,0:00:31.00,0:00:35.00,Default,,0000,0000,0000,,in the 2D case, our state was represented as an x and an ẋ--
Dialogue: 0,0:00:35.00,0:00:41.00,Default,,0000,0000,0000,,multiply some matrix by that state in such a way that we extract a measurement.
Dialogue: 0,0:00:41.00,0:00:48.00,Default,,0000,0000,0000,,In the 2D case the measurement was just x--just the x coordinate.
Dialogue: 0,0:00:48.00,0:00:52.00,Default,,0000,0000,0000,,We can think of this as a 1 x 1 vector or a 1 x 1 matrix.
Dialogue: 0,0:00:52.00,0:00:56.00,Default,,0000,0000,0000,,The matrix we use to do that was this one.
Dialogue: 0,0:00:56.00,0:01:01.00,Default,,0000,0000,0000,,That was our H matrix--1, 0--because 1 times x gives us the x,
Dialogue: 0,0:01:01.00,0:01:05.00,Default,,0000,0000,0000,,and 0 times ẋ gives us the nothing--exactly what we want.
Dialogue: 0,0:01:05.00,0:01:08.00,Default,,0000,0000,0000,,But now let's talk about the dimensionality of these matrices
Dialogue: 0,0:01:08.00,0:01:13.00,Default,,0000,0000,0000,,and how this multiplication yielded just this number x.
Dialogue: 0,0:01:13.00,0:01:20.00,Default,,0000,0000,0000,,So we can think of x here as a 1 x 1 matrix.
Dialogue: 0,0:01:20.00,0:01:24.00,Default,,0000,0000,0000,,We got that matrix by multiplying this one, which is a 1 x 2--
Dialogue: 0,0:01:24.00,0:01:31.00,Default,,0000,0000,0000,,one row by two columns--with this, which is two rows by one column.
Dialogue: 0,0:01:31.00,0:01:36.00,Default,,0000,0000,0000,,What we see here is that this 1 actually came from right here,
Dialogue: 0,0:01:36.00,0:01:40.00,Default,,0000,0000,0000,,and this 1 came from right here.
Dialogue: 0,0:01:40.00,0:01:43.00,Default,,0000,0000,0000,,These 2s we can think of as canceling out, in a way,
Dialogue: 0,0:01:43.00,0:01:46.00,Default,,0000,0000,0000,,giving us this 1 x 1 matrix.
Dialogue: 0,0:01:46.00,0:01:50.00,Default,,0000,0000,0000,,Now, let's see if we can generalize that to the 4-dimensional case as presented in the homework.
Dialogue: 0,0:01:50.00,0:01:58.00,Default,,0000,0000,0000,,In the 4-dimensional case our state is now given by x, y, ẋ, ẏ.
Dialogue: 0,0:01:58.00,0:02:00.00,Default,,0000,0000,0000,,We're going to have some H matrix.
Dialogue: 0,0:02:00.00,0:02:05.00,Default,,0000,0000,0000,,I don't know anything about it yet, but I'm just going to put this there for now as a placeholder.
Dialogue: 0,0:02:05.00,0:02:09.00,Default,,0000,0000,0000,,We want to get a measurement from that. What should this measurement be?
Dialogue: 0,0:02:09.00,0:02:13.00,Default,,0000,0000,0000,,It's not just going to be x, because now our position includes both x and y.
Dialogue: 0,0:02:13.00,0:02:19.00,Default,,0000,0000,0000,,So it's going to be a column vector--x and y.
Dialogue: 0,0:02:19.00,0:02:23.00,Default,,0000,0000,0000,,Again, let's think. What's going on with the dimensionality here?
Dialogue: 0,0:02:23.00,0:02:26.00,Default,,0000,0000,0000,,Here we have a 2 x 1 matrix,
Dialogue: 0,0:02:26.00,0:02:32.00,Default,,0000,0000,0000,,and that came from this matrix, which I said we don't know anything about yet--
Dialogue: 0,0:02:32.00,0:02:35.00,Default,,0000,0000,0000,,I'll just say a question mark by question mark--
Dialogue: 0,0:02:35.00,0:02:40.00,Default,,0000,0000,0000,,and this matrix, which is four rows by one column.
Dialogue: 0,0:02:40.00,0:02:44.00,Default,,0000,0000,0000,,Now, can you use the intuition we built up here
Dialogue: 0,0:02:44.00,0:02:51.00,Default,,0000,0000,0000,,for how the dimensionality of matrices works with this to fill in the question marks?
Dialogue: 0,0:02:51.00,0:02:56.00,Default,,0000,0000,0000,,Once you figure out the number of rows and the number of columns in this H matrix,
Dialogue: 0,0:02:56.00,0:02:59.00,Default,,0000,0000,0000,,figuring out where to put your 1s and 0s will be a little bit easier.
Dialogue: 0,0:02:59.00,9:59:59.99,Default,,0000,0000,0000,,I wish you luck.