Why Matrix Multiplication is Not Commutative: A Fundamental Principle in Linear Algebra

Matrix multiplication is a key concept in linear algebra, used extensively in various fields such as computer graphics, machine learning, and physics. One of the most important properties of matrix multiplication is its non-commutativity, meaning that the order in which matrices are multiplied can significantly affect the result. In this article, we will delve into why matrix multiplication is not commutative and explore the underlying principles, including the behavior of linear transformations and function compositions.

The Non-commutativity of Matrix Multiplication

Let's consider two matrices, A and B, as an example:

Example Matrices

A begin{bmatrix} 1 2 3 4 end{bmatrix}

B begin{bmatrix} 5 6 7 8 end{bmatrix}

When we multiply A and B, denoted as A * B, we get:

A * B begin{bmatrix} (1 * 5) (2 * 7) (1 * 6) (2 * 8) (3 * 5) (4 * 7) (3 * 6) (4 * 8) end{bmatrix}

Calculating the elements, we get:

A * B begin{bmatrix} (1 * 5) (2 * 7) (1 * 6) (2 * 8) (3 * 5) (4 * 7) (3 * 6) (4 * 8) end{bmatrix} begin{bmatrix} 19 22 43 50 end{bmatrix}

Now, let's compute B * A using the same matrices:

B * A begin{bmatrix} (5 * 1) (6 * 3) (5 * 2) (6 * 4) (7 * 1) (8 * 3) (7 * 2) (8 * 4) end{bmatrix}

Calculating the elements, we get:

B * A begin{bmatrix} (5 * 1) (6 * 3) (5 * 2) (6 * 4) (7 * 1) (8 * 3) (7 * 2) (8 * 4) end{bmatrix} begin{bmatrix} 23 34 31 46 end{bmatrix}

As we can see, A * B and B * A are not the same, which demonstrates that matrix multiplication is not commutative:

A * B ≠ B * A

Linear Transformations and Matrix Multiplication

In the context of linear algebra, matrix multiplication can be interpreted as the composition of linear transformations. To understand this, let's consider two linear transformations L and M. The composition of these transformations is denoted as L ° M or M ° L, depending on the order in which they are applied.

Composition of Linear Transformations

The composition L ° M means that the transformation M is applied first, followed by the transformation L. Similarly, M ° L means that L is applied first, followed by M. Mathematically, we can express this as:

L ° M (v) L (M (v)) and M ° L (v) M (L (v))

When these transformations are represented by matrices, the order in which they are multiplied affects the resulting matrix, just like the order in which the transformations are applied. This is why matrix multiplication is non-commutative.

Example of Transformation Composition

Consider the following linear transformations:

L: V → V and M: V → V

The composition of these transformations is denoted as L ° M or M ° L. For a vector v in the vector space V, we have:

(L ° M) (v) L (M (v)) and (M ° L) (v) M (L (v))

When the matrices representing these transformations are [L] and [M], the formulas for matrix multiplication need to be consistent with the composition of transformations. Therefore, the product [L][M] represents the composition L ° M, and [M][L] represents M ° L.

Function Composition and Commutativity

The behavior of function composition can be generalized to understand the non-commutativity of matrix multiplication. Function composition is the process of applying one function to the result of another. For two functions f and g, the composition (f ° g) is defined as:

(f ° g) (x) f (g (x))

Similarly, (g ° f) (x) g (f (x)). The commutativity of function composition means that the order in which the functions are applied does not matter, that is, (f ° g) (g ° f). However, in many cases, (f ° g) ≠ (g ° f).

For example, if f(x) x 1 and g(x) 2x, then:

(f ° g) (x) f (2x 1) 2x 2

(g ° f) (x) g (x 1) 2x 2

These two functions commute, but in general, function composition is not commutative. For instance, if f(x) x^2 and g(x) x 1, then:

(f ° g) (x) f (x 1) (x 1)^2 x^2 2x 1

(g ° f) (x) g (x^2) x^2 1

Clearly, (f ° g) ≠ (g ° f).

Conclusion

Matrix multiplication is non-commutative because the order in which linear transformations are applied affects the result. This property is deeply rooted in the nature of function composition, where the order of application generally matters. The fact that matrix multiplication encodes the composition of linear transformations makes non-commutativity a fundamental principle in linear algebra.

Understanding matrix multiplication and its properties is crucial for anyone working with linear algebra. By recognizing the non-commutativity of matrix multiplication, you can better grasp the behavior of linear transformations and how they interact with each other. Whether you're dealing with coordinate transformations in computer graphics, optimizing algorithms in machine learning, or studying the dynamics of physical systems, the principle of non-commutativity remains a constant and essential concept.