Now I'm going to give you a quick introduction to norms and Julia. Here I'm defining a vector. And the norm command is in the linear algebra package, which textbook package loads automatically. So all we have to do is say norm of x. And that is by default the same as the two norm. We also have definitions built-in for the one norm, which is the sum of elements in the absolute value. And the infinity norm, which is the maximum of the absolute value over all the elements. When it comes to matrix norms, it's very easy to make a mistake. Here I have a two-by-two matrix. And the normal command is defined for that matrix. But the way it works is that it just interprets the matrix as a vector again and just takes the sum of squares of all the elements in this than the square root. So that is a valid matrix norm, but it's not the one we usually use when we analyze algorithms. So instead we want to use op norm, op standing for operator because it's the interpretation of the norm for a as an operator. And so that would be what we call the matrix two norm. We can also do the matrix one norm. It turns out that that is equivalent to an easy formula. It's the maximum of the column sums. The one tells you to sum vertically. So if we sum the absolute value in the first dimension than the maximum of those two is the one norm of the matrix. And we can do the infinity norm of the matrix. And that's the maximum value of the sums across columns. Now unlike the one norm and the infinity norm, the 2-norm, doesn't it met a simple formula for calculating it. But it does have a really nice geometric interpretation that's useful to keep in mind. Sometimes. Here I'm going to define a bunch of angles on the unit circle. And then I'll just define vectors of points on that circle. So I'll do that by constructing a matrix whose first row are the x coordinates of the points and the second row or the Y coordinates of the points. Or in this case, we'd be thinking in them as the x1 coordinates and the x2 coordinates. When I do this, each column is a point on the unit circle. Whereas each row is one of those coordinates coming all the way around the circle. So if I plot these as points, you see we've filled up a lot of the circle. Now, since a times x is just a times each column of X, what this will do is it'll apply a to each column and store the result, which has also got length two in each column of y. So what we've done is we've computed the image of all of these points under the mapping. X goes to a times x. And when we look at those images, we get an ellipse. In higher dimensions, it would be an ellipsoid or a hyper ellipsoid. But that's what you always get when you apply a linear operator with a matrix to the points on the unit circle. And that two norm of the matrix is the radius of the smallest circle that contains all of those points. In other words, it's the larger of the two radii of the ellipse.
Section 2.7
From Tobin Driscoll September 11, 2020
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