Eigenvalues and vectors

Hi team,

I am confused by this graph, I don’t understand how a matrix can be represented by 6 points, some are stretched by 2 and 3, and some are just stretched somewhere in between. I can’t find any intuition in this section.


Screenshot 2023-09-19 102125

The arrows show how each specific point in that grid is affected by applying the [[2 1][0 3]] matrix to them. It’s a linear transformation.

This example is discussed between time marks 0:12 and 0:27 of the “Eigenvalues and Eigenvectors” video.

But the method is explained in the previous video “Eigenbases”.
Here is a screen capture from that video:

For each point in the grid, consider it to be the end point of a vector that starts at the origin.
So the eight points correspond to these eight vectors, going clockwise starting from the upper left point - note these are all column vectors, so you can compute (W * v) as a dot product:
[-1 1]
[0 1]
[1 1]
[1 0]
[1 -1]
[0 -1]
[-1 -1]
[-1 0]

For each of those vectors, you multiply it by the W = [[2 1][0 3]] matrix (as shown in the Eigenbases video), and you can compute the results for all eight points.

Here is an example for the first point (the upper left):
W * v = [[2 1] [0 3] * [-1 1] = [-1 3].
The end of the arrow drawn for the upper-left point is at coordinate (-1,3) in the “Finding eigenvalues” image.

1 Like

thanks a lot Tom, appreciate