Composition and Matrix Multiplication Composition and Matrix MultiplicationThe composition of two linear transformations is also linear. The geometrical reason is clear: if parallelograms and scalings are preserved by each transformation, then they are preserved by the combined transformation. See here for a rigorous proof.
Since linear transformations between euclidean spaces are equivalent to matrices, the composition of linear transformations should have corresponding operation on matrices.
Let A and B be the matrices for linear transformations T: Rn → Rm and S: Rk → Rn. Then the matrix for the composition TS: Rk → Rm is the product AB of A and B
Example Consider linear transformations
T: R2 → R2,
(y1, y2) →
(z1, z2) =
(a11y1 + a12y2,
a21y1 + a22y2)
S: R2 → R2,
(x1, x2) →
(y1, y2) =
(b11x1 + b12x2,
b21x1 + b22x2)
By substituting the formulae of y1, y2 in terms of x1, x2 into the formulae of z1, z2 in terms of y1, y2, we find the composition TS: R2 → R2, (x1, x2) → (z1, z2) is given by
z1
= a11(b11x1 + b12x2)
+ a12(b21x1 + b22x2)
= (a11b11 + a12b21)x1
+ (a11b12 + a12b22)x2
z2
= a21(b11x1 + b12x2)
+ a22(b21x1 + b22x2)
= (a21b11 + a22b21)x1
+ (a21b12 + a22b22)x2
In terms of the product of matrices, we have
| [ | a11 | a12 | ][ | b11 | b12 | ] = [ | a11b11 + a12b21 | a11b12 + a12b22 | ] |
| a21 | a22 | b21 | b22 | a21b11 + a22b21 | a21b12 + a22b22 |
Note that in the example above, the first entry a11b11 + a12b21 of the matrix AB is obtained by "multiplying" the first row (a11 a12) of A with the first column (b11 b21) of B. Similar observation can be made to the other entries. In fact, we have the following general rule for multiplying an m by n matrix A and an m by k matrix B together.
The result AB is an m by k matrix. The rigorous proof of the rule can be found in the next section.
| A = [ | 1 | 2 | ] |
| 3 | 4 |
| B = [ | -1 | 0 | 2 | ] |
| 0 | 1 | 3 |
| C = [ | 0 | 1 | -1 | ] |
| 2 | 0 | 3 | ||
| -1 | 1 | 2 |
| D = [ | 1 | 2 | ] |
| 2 | 3 | ||
| 3 | 4 |
Then
| AB = [ | 1×(-1) + 2×0 | 1×0 + 2×1 | 1×2 + 2×3 | ] = [ | -1 | 2 | 8 | ] |
| 3×(-1) + 4×0 | 3×0 + 4×1 | 3×2 + 4×3 | -3 | 4 | 18 |
| A2 = AA = [ | 1×1 + 2×3 | 1×2 + 2×4 | ] = [ | 7 | 10 | ] |
| 3×1 + 4×3 | 3×2 + 4×4 | 15 | 22 |
| DA = [ | 1×1 + 2×3 | 1×2 + 2×4 | ] = [ | 7 | 10 | ] |
| 2×1 + 3×3 | 2×2 + 3×4 | 11 | 16 | |||
| 3×1 + 4×3 | 3×2 + 4×4 | 15 | 22 |
| BC = [ | (-1)×0 + 0×2 + 2×(-1) | (-1)×1 + 0×0 + 2×1 | (-1)×(-1) + 0×3 + 2×2 | ] = [ | -2 | 1 | 5 | ] |
| 0×0 + 1×2 + 3×(-1) | 0×1 + 1×0 + 3×1 | 0×(-1) + 1×3 + 3×2 | -1 | 3 | 9 |
| (AB)C = [ | (-1)×0 + 2×2 + 8×(-1) | (-1)×1 + 2×0 + 8×1 | (-1)×(-1) + 2×3 + 8×2 | ] = [ | -4 | 7 | 23 | ] |
| (-3)×0 + 4×2 + 18×(-1) | (-3)×1 + 4×0 + 18×1 | (-3)×(-1) + 4×3 + 18×2 | -10 | 15 | 51 |
| A(BC) = [ | 1×(-2) + 2×(-1) | 1×1 + 2×3 | 1×5 + 2×9 | ] = [ | -4 | 7 | 23 | ] |
| 3×(-2) + 4×(-1) | 3×1 + 4×3 | 3×5 + 4×9 | -10 | 15 | 51 |
On the other hand, the products such as AD, AC, BA, CB, B2 are meaningless.
Example According to an earlier example, the matrix for the rotation by angle α is
| Rα = [ | cosα | - sinα | ] |
| sinα | cosα |
Since rotation by β and then rotation by α is the same as rotation by α + β, we must have RαRβ = Rα+β, i.e.,
| [ | cosα | - sinα | ] [ | cosβ | - sinβ | ] = [ | cos(α + β) | - sin(α + β) | ] |
| sinα | cosα | sinβ | cosβ | sin(α + β) | cos(α + β) |
If we compute matrix multiplication on the left side in the usual way, we get the following familiar formulae in trigonometry.
cosαcosβ - sinαsinβ = cos(α + β)
sinαcosβ + cosαsinβ = sin(α + β)