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\centerline{Mathematics 172 -- Numerical Analysis II}
\centerline{Discussion 3 -- ``How Big Is This Vector? How Big Is This Matrix?''}
\centerline{March 18, 1999}
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\noindent
{\bf Background}
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To understand the ``bad'' behavior of some of the systems  $Ax = b$
we have seen when floating point arithmetic is used with Gaussian Elimination, 
and to have tools at our disposal for analyzing 
other methods for solving systems, we will consider next the 
question:  How we can measure how close two matrices are to 
each other, or measure the ``size'' of a matrix.  In this context, 
by ``size'' we will {\it not\/}
mean the number of rows and columns; instead given an $n$, we will want to 
be able to compare any two $n \times n$ matrices $A$ and $B$, determine
which is ``bigger,'' determine how close $A$ is to $B$ and so on.

For {\it vectors\/} in ${\bf R}^n$ you should recall that similar
questions can be addressed by using the usual ``vector norm:''
$$\|\vec{x}\| = \|(x_1, x_2, \ldots, x_n)\| = \sqrt{x_1^2 + x_2^2 + \cdots + x_n^2}\leqno(1)$$
The size of the number $\|\vec{x}\|$ gives us a measure of the ``size''
of $\vec{x}$ (its geometric length), and the distance between 
$\vec{x}$ and $\vec{y}$ can be determined by finding $\|\vec{x} - \vec{y}\|$.  
From now on, we'll drop the ``arrow'' notation for vectors.
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\noindent
{\bf Discussion Questions}
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\noindent
Let's start with vectors and consider (1) in more detail.
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\item{A)}  What are the important properties of (1)?  In particular, 
\itemitem{1)}  What kind of numbers are norms of vectors?  What vectors
satisfy $\|x\| = 0$?
\itemitem{2)}  What happens to the norm if you multiply $x$ by a 
scalar $\lambda$?  How are $\|\lambda x\|$ and $\|x\|$ related?
\itemitem{3)}  How does (1) interact with {\it linear combinations\/}
of vectors?  (For instance, what can you say about $\|x + y\|$?)
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\noindent
For our purposes, it will be best to think of (1) as defining
a function $n : {\bf R}^n \to{\bf R}$, where $n(x) = \|x\| = 
\sqrt{\sum_{i=1}^n x_i^2}$.
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\item{B)}  Which (if any) of the following functions from
${\bf R}^n$ to ${\bf R}$ satisfy the same properties as (1)?
Explain.
\itemitem{1)} $n_1(x) = \root 3 \of {\sum_{i=1}^n x_i^3}$.   
\itemitem{2)} $n_2(x) = |x_1|\cdot |x_2| \cdot \cdots \cdot |x_n|$.
\itemitem{3)} $n_3(x) = |x_1| + |x_2| + \cdots + |x_n|$.
\itemitem{4)} $n_4(x) = \max \{|x_i|, i = 1,\ldots, n\}$  (the largest component
of the vector in absolute value).
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\item{C)}  For each of the functions $n_j$ above that {\it does\/}
have properties similar to the usual vector norm, draw a picture
of the set of vectors $x \in {\bf R}^2$ satisfying $n_j(x) = 1$
(analogues of the unit circle for (1)!)
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\noindent
One possible way to measure the ``size'' of a matrix $A$ 
(thinking of it as a mapping) is
{\it to measure how it affects the sizes of vectors, when you multiply 
$A$ times them\/}, for instance by considering the ratios
$$r_A(x) = {\|Ax\| \over \|x\|}$$
for $x \ne 0$.  If $A$ magnifies the lengths of vectors a lot, 
we will think of it as ``big''.  If $A$ shrinks the lengths of vectors a lot, 
we will think of it as ``small''.
For this purpose, {\it any\/} vector norm 
on ${\bf R}^n$ can be used (that is any function satisfying 
the same properties as the usual vector norm, as in question A).
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\item{D)}  How are $r_A(x)$ and $r_A(\lambda x)$ related?  What
does this tell you?  (Note, for any non-zero vector $x$, there is a 
vector of norm 1 (a ``unit vector'') in the same direction: 
$$u = {1\over \|x\|} x$$
Is it necessary to consider {\it all vectors\/} to study the possible
values of $r_A(x)$?)
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\item{E)}  What is the {\it biggest possible value\/} you 
can get for $r_A(x)$ for 
$$A = \pmatrix{7&1\cr
               -1&3\cr}$$
\itemitem{1)} using the vector $\ell_1$ norm on ${\bf R}^2$:
$\|(x,y)\|_1 = |x| + |y|$?
\itemitem{2)} using the vector $\ell_\infty$ norm on ${\bf R}^2$:
$\|(x,y)\|_\infty = \max\{|x|,|y|\}$?
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\itemitem{3)}  Do you notice any patterns here?  Can you 
determine what the biggest $r_A(x)$ will be for a general 
$2 \times 2$ matrix
$A$, using the norms from 1 and 2.
\itemitem{4)}  Recall that an {\it eigenvalue\/} of a matrix 
$A$ is a number $\lambda$ such that there exists a {\it non-zero\/}
vector $x$ satisfying an equation $Ax = \lambda x$.  How do 
the eigenvalues of $A$ relate to the maximum of the quotients $r_A(x)$?   
Is there an equality you can state?  An {\it inequality\/}?
\bye 