In higher dimensions, say, k, singular matrices can squish space into k-1, k-2, …, or 0 dimensions. The matrix A = (0, 0 \ 0, 0) would squish the space down to a point, namely (0 0).
Actually, an infinite number do, and we don’t know which you started with.Ī = (2, 3 \ 2, 3) squished the space down to a line. Several different x values get squished into that same value of y. It is because the space is so squished after transformation by y = Ax that one cannot take the resulting y and get back the original x. It means this:Ī singular matrix A compresses the space so much that the poor space is squished until it is nothing more than a line.
Okay, tell me what it means for a matrix to be singular. Well, you are no doubt thinking, this is all very entertaining. In a like manner, 3 x 3 matrices do the same to 3-space 4 x 4 matrices, to 4-space and so on. They skew,stretch, compress, rotate, and even flip 2-space.
In the transformed space, the corresponding triangle ends up at the bottom right! A = (1, 2 \ 3, 1) appears to be an innocuous matrix - it does not even have a negative number in it - and yet somehow, it twisted the space horribly. In the original grid, the triangle is located at the top left. This matrix flips the space! Notice the little triangles. Here’s an interesting matrix that produces a surprising result: A = (1, 2 \ 3, 1). Note the location of the triangle this space was rotated around the origin. The original grid, I’m showing in red the transformed grid is shown in blue.Ī pure rotation (and stretching) looks like this: We need first to become familiar with pictures like this, so let’s see some examples. I will graph the region that makes it easiest to see the point I wish to make, but you must remember that whatever I’m showing you applies to the entire space. Regardless of the region graphed, you are supposed to imagine two infinite planes. I could just as well have transformed a wider area. I’ve suppressed the scale information in the graph, but the axes make it obvious that we are looking at the first quadrant in the graph above. They wouldn’t be necessary had I transformed a picture of the Eiffel tower. They are there to help you orient the transformed grid relative to the original. I put a triangle and circle at the bottom left and top left of the original grid, and then again at the corresponding points on the transformed grid. Notice that in the above image there are two small triangles and two small circles. The distorted image might not be helpful in understanding the Eiffel Tower, but it is helpful in understanding the properties of A. The result would be a distorted version of the original image, just as the the grid above is a distorted version of the original grid. I used a grid above, but I could just as well have used a picture of the Eiffel tower and, pixel by pixel, transformed it by using y = Ax. I want you to think about transforms like A as transforms of the space, not of the individual points. In this way, I can now see exactly what A = (2, 1 \ 1.5, 2) does. Then I’m going to graph the transformed points:įinally, I’m going to superimpose the two graphs: One at a time, I’m going to take every point on the grid, call the point x, and run it through the transform y = Ax. To do that, I’m first going to take a grid, To focus better on A, we are going to graph y = Ax for all x. I do not want you to get lost among the individual points which A could transform, however. To get a better understanding of how A transforms the space, we could graph additional points: That is, we are going to think about A in terms of its effect in transforming points in space from x to y. Where commas are used to separate elements on the same row and backslashes are used to separate the rows. If you had more imagination, we could use the technique on 4 x 4, 5 x 5, and even higher-dimensional matrices.īut we will limit ourselves to 2 x 2. The technique I’m about to show you could be used with 3 x 3 matrices if you had a better 3-dimensional monitor, and as will be revealed, it could be used on 3 x 2 and 2 x 3 matrices, too. That will be, as they say, without loss of generality. I’m going to show you a way of graphing square matrices, although we will have to limit ourselves to the 2 x 2 case. The topic for today is the square matrix, which we will call A. I want to show you a way of picturing and thinking about matrices.
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