Matrices and other generalities

As stated above we cannot always find some matrix such that . As a simple example, consider the cases of

> A:=matrix(2,2,[0$4]);

> A:=matrix(2,2,[1,2,1,2]);

Why is the above matrix an example?

Later we will learn how to test for the existence of and find it if needed.

In any case, the whole procedure is reducing to writing a suitable matrix and manipulating it by elementary operations or equivalently by multiplying by other matrices.

We study a variation of matrix manipulation which is useful for understanding and some calculations. So far, we have studied matrices whose entries are numbers, but we can think of them as matrices provided we are very careful. We should make sure that all matrices in any row or column have comparable sizes and when we multiply two such super-matrices, we should respect the order of multilication.

The general advice is that don't use such super-matrices unless you understand them well and are very careful.

Examples:

> G:=matrix(2,2,[u1,u2,u3,u4]);`G^2`:=evalm(G&*G);

>

If are matrices, then this has to be evaluated very carefully. Note the order in which Maple writes the product in the above expression. If we switch the orders, the answer does not work as we illustrate below.

In particular, note that cannot be factored as . Check this.

> u1:=matrix(2,2,[1,0,1,1]);u2:=matrix(2,2,[0,1,1,0]);
u3:=matrix(2,2,[1,2,3,4]);u4:=matrix(2,2,[1,3,2,5]);

> GG:=blockmatrix(2,2,[u1,u2,u3,u4]);

> evalm(GG^2);

The next line shows a correct evaluation of the square matrix.

> blockmatrix(2,2,[(u1^2+u2&*u3),(u1&*u2+u2&*u4),
(u3&*u1+u4&*u3),(u3&*u2+u4^2)]);

The next line shows an incorrect evaluation of the square matrix.

> blockmatrix(2,2,[(u1^2+u2&*u3),(u1&*u2+u2&*u4),(u1&*u3+u3&*u4)
,(u2&*u3+u4^2)]);

You should try other variations.

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