Introduction: Chaining Transformations
When you apply one transformation after another, the combined effect is also a linear transformation. Matrix multiplication is how we compute the matrix of the combined transformation. This is why matrix multiplication has its peculiar "row times column" rule.
The Matrix Multiplication Rule
If \(A\) is \(m \times n\) and \(B\) is \(n \times p\), then \(C = AB\) is \(m \times p\) with entries:
\[c_{ij} = \sum_{k=1}^{n} a_{ik} b_{kj}\]Each entry is the dot product of row \(i\) of \(A\) with column \(j\) of \(B\).
Key insight: \((AB)\vec{v} = A(B\vec{v})\)—applying \(B\) first, then \(A\).
Properties of Matrix Multiplication
- Associativity: \((AB)C = A(BC)\)
- Distributivity: \(A(B + C) = AB + AC\)
- NOT Commutative: In general, \(AB \neq BA\)
Worked Examples
Example 1: Basic Multiplication
\[A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}, \quad B = \begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix}\] \[AB = \begin{pmatrix} 1 \cdot 5 + 2 \cdot 7 & 1 \cdot 6 + 2 \cdot 8 \\ 3 \cdot 5 + 4 \cdot 7 & 3 \cdot 6 + 4 \cdot 8 \end{pmatrix} = \begin{pmatrix} 19 & 22 \\ 43 & 50 \end{pmatrix}\]Example 2: Non-Commutativity
Using the same \(A\) and \(B\):
\[BA = \begin{pmatrix} 5 \cdot 1 + 6 \cdot 3 & 5 \cdot 2 + 6 \cdot 4 \\ 7 \cdot 1 + 8 \cdot 3 & 7 \cdot 2 + 8 \cdot 4 \end{pmatrix} = \begin{pmatrix} 23 & 34 \\ 31 & 46 \end{pmatrix}\]Clearly \(AB \neq BA\)!
Example 3: Composing Transformations
Rotate 90° then reflect across x-axis:
\[R = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}, \quad M = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}\] \[MR = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} = \begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}\]This is reflection across the line \(y = -x\).
The Quantum Connection
The non-commutativity of matrix multiplication is the mathematical origin of the uncertainty principle. When \(\hat{A}\hat{B} \neq \hat{B}\hat{A}\), the observables cannot be simultaneously measured with arbitrary precision. The commutator \([\hat{A}, \hat{B}] = \hat{A}\hat{B} - \hat{B}\hat{A}\) quantifies this incompatibility.