A Minimum Length Solution Example
In the example I have been using in the last few classes,
the
of the solution vector in terms of free variables z and w is 
, i.e.
, assuming I did my algebra right.
We’ll study functions like this in Chapter 6, but for just finding the minimum all you do is find the partial derivatives, set them both equal to zero, and solve the resulting system of two linear equations in two variables. Explicitly
. We can write this in
matrix form as 
, and then solve by any of the many techniques you now
know. [Note that the matrix in this
system is symmetric. This always happens
when you’re minimizing a quadratic function of several variables. Why? See Chapter 6.]
Although in general the strategy of finding the inverse of
the matrix and multiplying through by it is usually the second-worst of the
many alternatives, for 2x2’s it’s sometimes not so bad. Explicitly:
.
Now what this means is that these values of z and w minimize the length of the whole solution 
. So we should
say that the minimum-length solution of
the original system of equations is 
. [Apologies if I got a couple of numbers wrong! Let me know.
Thanks.]
To reiterate, the idea of finding a minimum-length solution
is to go from a situation in which there is an infinite number of solutions and
you don’t have any idea which to choose, to a situation in which you pick out
one with some extra special property.
When this situation presents itself in the real world, that there is
more than one solution to a given problem, we often seek the solution that
minimizes cost or time or some other kind of unpleasantness. Whatever it is, you can often apply the same
technique of setting partial derivatives equal to zero and solving the
resulting equations. The minimum-length
idea is a simple example of this kind of problem, and it does have some nice
special properties arising from its geometric meaning.