In the normal manual method, you have to keep track of which row and column you're at in both matrices. If you forget, refinding your place can be difficult. There is an easier method that I call the proportions-vectors method :
+- -+
+- -+ | 5 6 |
| 0 5 6 11 13 | | 11 -5 |
| 10 -1 8 0 0 | x | 0 2 |
| 0 0 1 0 6 | | 0 0 |
+- -+ | 1 0 |
+- -+
Assume you have a list of vectors that you want to mix together in certain proportions. The left matrix has the proportions for three mixes. The right matrix has the five vectors that are going to be squeezed together in various ways :
+- -+
+- -+ | ( 5 6) |
| 0 5 6 11 13 | | (11 -5) |
| 10 -1 8 0 0 | x | ( 0 2) |
| 0 0 1 0 6 | | ( 0 0) |
+- -+ | ( 1 0) |
+- -+
Now multiply :
+- -+
| 0(5 6) + 5(11 -5) + 6(0 2) + 11(0 0) + 13(1 0) |
| 10(5 6) - 1(11 -5) + 8(0 2) + 0(0 0) + 0(1 0) |
| 0(5 6) + 0(11 -5) + 1(0 2) + 0(0 0) + 6(1 0) |
+- -+
Simplify each mix :
+- -+
| (68 -13) |
| (61 71) |
| ( 6 2) |
+- -+
Remove vector notation :
+- -+
| 68 -13 |
| 61 71 |
| 6 2 |
+- -+
Now you only have to pay attention to which row you're at in both matrices. If you forget, refinding your place is much easier.
Not only that, but figuring out other things about matrices is much easier :
- Matrix multiplication isn't commutative because you can't expect to exchange the proportions and the vectors and get the same result (except in special cases).
- The number of columns on the left has to match the number of rows on the right because both tell you how many vectors you're squeezing together.
- The number of rows on the left matches the number of rows in the result because both tell you how many different mixes you made.
- The number of columns on the right matches the number of columns in the result because both tell you the dimension of the vectors.