Introduction: Matrices Are Functions
A matrix is not just a table of numbers—it's a function that transforms vectors. When you multiply a matrix by a vector, you're sending that vector to a new location in the vector space. Understanding matrices as transformations is the key to understanding quantum operators.
Linear Transformations
A function \(T: V \to W\) is a linear transformation if:
- \(T(\vec{u} + \vec{v}) = T(\vec{u}) + T(\vec{v})\)
- \(T(c\vec{v}) = cT(\vec{v})\)
Every linear transformation between finite-dimensional spaces can be represented by a matrix, and every matrix defines a linear transformation.
Matrix-Vector Multiplication
If \(A\) is an \(m \times n\) matrix and \(\vec{v}\) is an \(n \times 1\) column vector:
\[A\vec{v} = \begin{pmatrix} a_{11} & a_{12} & \cdots \\ a_{21} & a_{22} & \cdots \\ \vdots & \vdots & \ddots \end{pmatrix} \begin{pmatrix} v_1 \\ v_2 \\ \vdots \end{pmatrix}\]Each entry of the result is a dot product of a row of \(A\) with \(\vec{v}\).
Worked Examples
Example 1: Rotation in 2D
Rotation by angle \(\theta\) counterclockwise:
\[R_\theta = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}\]Apply to \(\vec{v} = (1, 0)\) with \(\theta = 90°\):
\[R_{90°} \begin{pmatrix} 1 \\ 0 \end{pmatrix} = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} 1 \\ 0 \end{pmatrix} = \begin{pmatrix} 0 \\ 1 \end{pmatrix}\]The vector is rotated from pointing right to pointing up.
Example 2: Scaling
Scaling by factors \(s_x\) and \(s_y\):
\[S = \begin{pmatrix} s_x & 0 \\ 0 & s_y \end{pmatrix}\]Apply \(S = \begin{pmatrix} 2 & 0 \\ 0 & 3 \end{pmatrix}\) to \(\vec{v} = (1, 1)\):
\[S\vec{v} = \begin{pmatrix} 2 \\ 3 \end{pmatrix}\]Example 3: Reflection
Reflection across the x-axis:
\[M = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}\]Apply to \(\vec{v} = (2, 3)\): \(M\vec{v} = (2, -3)\).
The Quantum Connection
In quantum mechanics, operators act on state vectors just as matrices act on ordinary vectors. The Hamiltonian \(\hat{H}\) transforms states according to energy. Measurement operators project states onto eigenspaces. The entire dynamical evolution of a quantum system is a sequence of linear transformations.