VECTOR AND MATRIX NORMS

The question of “how big?” is a vector is not very complicated, and there’s no reason not to do it early on in a course like this. In fact, when you took Vector Calculus (Math 223 or some equivalent), one of the first things you did was learn the Pythagorean formula for the length of a vector. We saw that we can express the same thing in terms of matrix multiplication, which applies to vectors with any number of components.

There’s actually a variety of ways you can “size up” a vector, and we call them all norms. Specifically, a norm is a measure that satisfies three properties: (1)Nonnegativity: for any vector , with equality if and only if , the vector of all 0’s. (2)Scaling: for any scalar (regular number) . (3)Triangle inequality: for any vectors and . These make intuitive sense for Euclidean geometry, but they also hold for other ways to measure vectors. In class I gave you the “max” norm: maximum of the absolute values of the elements of . (These are instances of the p-norm which gives the Euclidean norm when and the max norm in the limit . But you don’t have to know that in Math410!)

What about the question of “how big” is a matrix? It would be really nice if there was a measure that had some connection with the operation of multiplying matrices by vectors to get other vectors. We make it so by defining it that way: if you have any norm defined for vectors, you can define it for matrices by over all possible nonzero vectors . This automatically gives us the great property, the Multiplicative Triangle Inequality, that by definition for any. In fact for any pair of matrices A, B for which the product is defined. It also turns out that the regular triangle inequality automatically follows from this definition.

The only minor snag is that this definition says nothing about how to actually CALCULATE ! This can be very tricky, and we’ll have to wait until Chapter 6 to learn how to do so for the Euclidean norm. For the max norm it’s pretty straightforward. You first sum the absolute values of the elements in each row, obtaining a column vector of nonnegative numbers. Then you take the max of the elements of that vector, and that is the max norm of the matrix.

For example, if we’re given the matrices , the sum of absolute values of elements in each row of A is , of which the maximum is 15. So . Similarly,

Let’s do a fast illustration of the matrix triangle inequalities I mentioned above. First, if we “randomly” choose the vector Sure enough, . As another illustration, use . This was not randomly chosen, but cooked up to make the triangle inequality an equality. Do you see how I went about cooking it up? Think about it.

Moreover you can quickly check (in fact, please do – I did the calculations rather hurriedly) that

Again, , and .