Matrix operations, a jump ahead

We begin by formally defining operations of matrices which will permit our equations to be meaningful. As stated above, our equations are to be thought of as where, in our example, we have

> A:=matrix(3,3,[0,1,2,1,-1,3,2,1,11]),X=matrix(3,1,[x,y,z]),B=matrix(3,1,[1,3,1]);

Clearly, we want the quantity to denote the column of the three left hand sides of the equations. This suggests that we define the matrix times the column as obtained by multiplying each row of by the column and stacking up the answers. Further each row times the column is obtained by multiplication of corresponding entries and adding up the result. Thus

More generally a matrix and a matrix can be multiplied to produce where each entry in the -th row and -th column of is produced by multiplying the -th row of with the -th column of .

It is essential that the rows of and columns of have the same sizes. For example:

> U:=matrix(3,2,[1,2,3,4,5,6]):V:=matrix(2,2,[a,b,c,d]):
op(U),op(V),evalm(U&*V);

Here is a formal precise definition. Let and be the entries of the matrices in the -position. Then the entry in the -position of the product can be defined as where it is understood that has columns and has rows. The entries give the -th row of as runs from to , while the entries , give the -th column of .

There is another way of thinking of the product which is very useful. The matrix can be thought of as three columns augmented together and each is a coefficient of one of the variables. So the equations become:

> A:=matrix(3,3,[0,1,2,1,-1,3,2,1,11]):B=matrix(3,1,[1,3,1]):

> x*submatrix(A,1..3,[1])+y*submatrix(A,1..3,[2])+z*submatrix(A,1..3,[3])=B;

>

Thus in general, we can think of the product as giving the various combinations columns of where the coefficients of each combination come from the entries of columns of .

A similar explanation can be given as combinations of rows of stacked together. Compare the example above.

All these views have useful applications.

The column combinations noted above, suggest a natural way to add matrices, namely add two matrices together, if they have the same sizes. The rule shall be to simply add the corresponding entries and arrange in the same shape.

Also, if we wish to multiply a matrix by a single number, it seems natural to multiply all entries by the same number. Thus, for example:

> L1:=matrix(2,2,[1,2,3,4]);L2:=matrix(2,2,[4,3,2,1]);

> `L1+L2`=evalm(L1+L2),`3L1`=evalm(3*L1),`2L1-3L2`=evalm(2*L1-3*L2);

>