Theoretical Principles & Computational Physics Guide
Matrix Representation: 2D Arrays & Multi-Dimensional Data Layout
Dr. Alorika Chatterjee & Dr. Anirban Shaw
Science & Physics Learners
Python 3 / NumPy / SciPy
1. Core Principles & Theoretical Framework
Linear Operators: Matrices represent linear transformations, coordinate rotations, and quantum Hamiltonian operators.
2D Array Indexing: Structured as nested lists or contiguous memory blocks indexed by row and column: $A_{ij} = \texttt{A[i][j]}$.
Memory Row-Major: In standard C/Python layout, elements along a row occupy adjacent memory addresses.
2. Mathematical Formulation & Algorithmic Pipeline
General $m \times n$ matrix index mapping:
$$ \mathbf{A} = \begin{pmatrix} a_{11} & a_{12} & \cdots & a_{1n} \ a_{21} & a_{22} & \cdots & a_{2n} \ \vdots & \vdots & \ddots & \vdots \ a_{m1} & a_{m2} & \cdots & a_{mn} \end{pmatrix}, \quad \text{1D Flat Index: } k = i \cdot n + j $$
3. Numerical Mechanics & Implementation Logic
- Nested Allocation: Instantiate using nested lists:
[[0.0 for _ in range(cols)] for _ in range(rows)]. - Row-Col Traversal: Traverse outer index $i$ for rows and inner index $j$ for columns to preserve cache locality.
- Dimension Verification: Validate that row lengths are homogeneous before performing linear algebraic operations.
4. Relevant Computational Practice Exercises
Exercise 1
Construct a $3 \times 3$ identity matrix and an arbitrary coordinate rotation matrix for angle $\theta = 45^\circ$.
Exercise 2
Write a function that verifies whether an input 2D list is a valid rectangular matrix.
Exercise 3
Flatten a $3 \times 3$ matrix into a 1D array of 9 elements and reconstruct it back.
- Gilbert Strang, "Linear Algebra and Its Applications", Cengage Learning (4th ed., 2006).