6.11 - More Matrix Math Concepts¶

Order Matters!¶

In mathematics, if a series of operations can be performed in any order, we say the operations are “commutative.” For example, the multiplication of numbers is commutative because the multiplications can be done in any order.

// These 3 statements calculate the same answer.
// Multiplication of numbers is commutative.
a = 3 * 4 * 7 * 10;
b = 10 * 7 * 4 * 3;
c = 4 * 10 * 7 * 3;

However, in matrix algebra, multiplication is NOT commutative. The order of the terms matter. If you change the order, you get a different result. For example:

Matrix Multiplication is Associative¶

In mathematics, if a series of operations can be grouped in different ways and you still get the same result, the operations are “associative.” For example, multiplication of numbers is associative because the order in which you perform the operations does not matter.

// These 3 statements calculate the same answer.
// Multiplication of numbers is associative.
a = (3 * 4) * (7 * 10);
b = 3 * ((4 * 7) * 10);
c = ((3 * 4) * 7) * 10;

In matrix algebra, multiplication is associative! This means that no matter which multiplications you perform first, second, or third, you get the same result. For the following three example equations, if you performed the multiplication in the parentheses first, you get the same result for T.

The fact that multiplication of matrices is associative is what allows us to combine a series of transformations into a single, 4-by-4 transformation matrix. We could multiply a vertex by a series of individual matrices, but this would be computationally expensive. We get a huge advantage in rendering speed if we combine all of the transformations into a single transform.

Transformations Are Relative¶

Conceptually, a programmer must keep transformations separate because each transform is actually processing a different model! Let’s explain that with an example. Suppose you have a model that is being transformed by five different matrices like this:

Conceptually, the matrix T5 is applied first to the model and now the model has different values for each of its vertices. The matrix T4 is now transforming the transformed model. This logic applies to each of the succeeding multiplications. To make this idea more concrete, suppose that transform T5 was a scaling operation that is changing the model’s vertices from units of feet to inches. T5 would be this scaling matrix:

Since T5 has changed the conceptual units of the model, all of the other transforms must conceptually think in terms of feet, instead of in inches, because they are now acting on the transformed model!

In summary, when you create a transformation matrix, you must consider the transformations that have come before it. We will see more examples of this when we study projection transforms.

Glossary¶

commutative property

the order that a math operation is performed does not affect the result. (e.g., 3*7 == 7*3). Matrix multiplication is NOT commutative.

associative property

the grouping of math operations does not affect the result of a calculation. (e.g. (3 + 4) + 5 == 3 + (4 + 5)). Matrix multiplication is associative.

Self Assessment¶