Matrix Norms

Now we consider measuring the size of an $n \times n$ matrix. We can use much of what we developed for vectors. Notice that the definition of a vector norm was tied to the algebraic operations for vectors. We noted what happened to our measure of size in the presence of the scalar multiplication operation namely

$$\Vert \alpha \mathbf{x}\Vert = \vert\alpha\vert \Vert \mathbf{x}\Vert.$$

This explains how the size of a vector scales. We also discussed the effects of vector addition in the triangle equality

$$\Vert\mathbf{x}+ \mathbf{y}\Vert \leq \Vert \mathbf{x}\Vert + \Vert \mathbf{y}\Vert.$$

We mention these because matrices has a larger set of algebraic operations associated with it. As with vectors, we can add matrices and multiple them by scalars. We can also multiply two matrices together to produce another matrix, which was something we failed to do with vectors. The property chosen to address this operation is

$$\Vert \mathbf{A}\mathbf{B}\Vert \leq \Vert \mathbf{A}\Vert\Vert \mathbf{B}\Vert.$$

A similar relationship was chosen for the mixed case of a vector and matrix multiple

$$\Vert \mathbf{A}\mathbf{x}\Vert _{v} \leq \Vert \mathbf{A}\Vert _{m} \Vert \mathbf{x}\Vert _{v}$$

where the subscript $v$ denotes the vector norm and $m$ denotes the matrix norm.

We are now ready to define the matrix norm

There is a bit of an ambiguity in property 5. We know that a vector space can have several norms associated with it. Does property 5 apply to any vector norm for a given matrix norm? In fact, it just says that the matrix norm needs to be compatible with a some vector norm. Let's turn the question around. Suppose you like a particular vector norm, which matrix norm satisfies the compatibility requirement?

To address this question, mathematicians constructed the operator norm associated with a given vector norm. What this means is that given a vector norm, they built an operator norm with all of the desired properties including (5).

Of course it must be shown that a this defines a matrix norm, which it does. This definition is built to make property 5 automatic.