Let A and Y be given matrices with the same number of rows. The solutions X of AX = Y are found in three steps: (1) Find unimodular matrices M and N such that MAN is diagonal, call it D. (2) Find all solutions Z of DZ = MY. (3) Set X = NZ.
You should first focus on the case in which A is diagonal. Once you understand this case, you will see that the solution procedure is just a method for reducing the general problem to this special case.
Step (1) is the difficult one in hand calculation, but with Matlab it is solved with a single command, [D,M,N] = quick(A), as was explained in Chapter 2.
You may want to work through the examples in Section 3 and in the Examples section, but it is also easy to create your own exercises in matrix division. Start with A and X, compute Y = AX, and then see whether you can recover X from a knowledge of just A and Y. The examples in the book were designed with hand calculation in mind and are therefore rather tame. See Example for an example that is more like what you might find if you just strike out on your own. Then stike out on your own.
If you choose as your matrix A the 3x3 matrix with rows [1 2 3], [4 5 6], and [7 8 9] you will get results very different from the ones you will get if your A is `random', because this matrix is a zero divisor.
If A is 2x2 or 4x4 instead of 3x3, the answers are essentially the same. When A is not a zero divisor, X can always be recovered when A and AX are known. The cases in which A is a zero divisor are more interesting. Try cases in which the diagonal matrix equivalent to A has more than one zero on the diagonal. There can be two zeros for a nonzero 3x3 matrix, and as many as 3 for a nonzero 4x4 matrix. Try all the cases and notice how the description of the solutions of AX = 0 depends on the number of zeros on the diagonal of D. Note also that it involves certain columns of N.
Try some examples in which the matrix A you are dividing by is wide and some in which it is high.
Try some problems of division on the right.
For a given A, it is easy to work out necessary and sufficient conditions on Y for the equation AX = Y to have solutions.
Perhaps the most basic problem of elementary linear algebra is the solution of an mxn system of inhomogeneous linear equations. This problem is just the solution of a problem of matrix division, that is, the solution of AX = Y, where A is a given mxn matrix and Y is mx1. The methods of this chapter assume A and Y have integer entries and gives all solutions X whose entries are integers. If rational number solutions are acceptable, the problem is easier in that divisibility considerations can be ignored (see Chapter 6).
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