22 Jul 2025

One- and Two-Dimensional Arrays in Python

python arrays one-dimensional-arrays two-dimensional-arrays

An array stores related values under one name and selects each element by an index. In introductory Python, lists provide the required one- and two-dimensional array structure without an additional library.

One-dimensional arrays

A one-dimensional array represents a vector or a sequence of sampled data:

positions = [0.0, 0.5, 1.0, 1.5]

Python indices begin at zero. Initialization, access, and update are therefore

n = 5
values = [0.0] * n

values[0] = 2.5
values[3] = 7.0

print(values[0])
print(values[3])

The valid indices are $0,1,\ldots,n-1$. A loop processes every element:

x = [0.0, 0.5, 1.0, 1.5]
y = [0.0] * len(x)

for i in range(len(x)):
    y[i] = x[i]**2

This stores the sampled values $y_i=x_i^2$.

Numerical application: a dot product

For vectors $\mathbf a,\mathbf b\in\mathbb R^n$,

\[\mathbf a\cdot\mathbf b=\sum_{i=0}^{n-1}a_i b_i.\]

The formula translates directly:

a = [1.0, 2.0, 3.0]
b = [4.0, 5.0, 6.0]

dot = 0.0
for i in range(len(a)):
    dot = dot + a[i] * b[i]

print(dot)             # 32.0

Both arrays must have the same length.

Two-dimensional arrays

A two-dimensional array represents a table or matrix. A list of row lists gives

A = [
    [4.0, 1.0, 1.0],
    [2.0, 5.0, 2.0],
    [1.0, 2.0, 4.0]
]

print(A[1][2])         # row 1, column 2: 2.0
A[0][0] = 5.0

An $m\times n$ zero matrix should be initialized with a separate row for each $i$:

rows = 3
columns = 4

A = [[0.0 for j in range(columns)] for i in range(rows)]

Nested loops visit every entry:

for i in range(rows):
    for j in range(columns):
        A[i][j] = i + j

Numerical application: matrix-vector multiplication

For

\[y_i=\sum_{j=0}^{n-1}A_{ij}x_j,\]

use one outer loop for the output row and one inner loop for its sum:

A = [
    [2.0, 1.0],
    [1.0, 2.0]
]
x = [1.0, -1.0]
y = [0.0] * len(A)

for i in range(len(A)):
    total = 0.0
    for j in range(len(x)):
        total = total + A[i][j] * x[j]
    y[i] = total

print(y)               # [1.0, -1.0]

The same indexing pattern appears in Gaussian elimination, Jacobi iteration, interpolation tables, and polynomial fitting. The dimensions of every array should be checked before such operations are performed.

The row-by-row initialization shown above also avoids aliasing. In contrast,

A = [[0.0] * columns] * rows

repeats references to one row object. Changing A[0][1] would then change column $1$ in every apparent row. A nested comprehension creates independent row lists and preserves the intended two-dimensional indexing.

Solved Problems

1. Mean and deviations stored in one-dimensional arrays

For

values = [2.0, 4.0, 5.0]

the mean is

\[\bar x=\frac{2+4+5}{3}=\frac{11}{3}.\]

A loop can form the deviations:

mean = sum(values) / len(values)
deviation = [0.0] * len(values)

for i in range(len(values)):
    deviation[i] = values[i] - mean

Thus

\[\mathbf d=\left(-\frac53,\frac13,\frac43\right).\]

Their sum is

\[\sum_i d_i=-\frac53+\frac13+\frac43=0,\]

as required by the definition of the arithmetic mean.

2. Matrix multiplication with nested loops

Let

\[A=\begin{pmatrix}1&2\\3&4\end{pmatrix},\qquad B=\begin{pmatrix}2&0\\1&2\end{pmatrix}.\]

The product entry is $C_{ij}=\sum_{k=0}^{1}A_{ik}B_{kj}$. Therefore

\[C_{00}=1(2)+2(1)=4,\] \[C_{01}=1(0)+2(2)=4,\] \[C_{10}=3(2)+4(1)=10,\] \[C_{11}=3(0)+4(2)=8.\]

Hence

\[\boxed{C=AB=\begin{pmatrix}4&4\\10&8\end{pmatrix}}.\]

The inner dimensions are both $2$; without this agreement, the sum over $k$ would not be defined.

Descriptive Questions

Explain zero-based indexing and state the valid indices of a one-dimensional array of length $n$.
Describe initialization, access, and update operations for one- and two-dimensional Python lists.
Translate the mathematical definitions of a dot product and matrix product into loop-index form.
Explain row aliasing in a repeated-list initialization and how a nested comprehension avoids it.

Numerical Problems

For values = [3, 6, 9, 12], state values[0] and values[-1]. Answer: 3 and 12.
Use nested loops to sum all entries of $\begin{pmatrix}1&2&3\4&5&6\end{pmatrix}$. Answer: $21$.
Compute $A\mathbf x$ for $A=\begin{pmatrix}1&-1\2&3\end{pmatrix}$ and $\mathbf x=(4,2)^T$. Answer: $A\mathbf x=(2,14)^T$.
Execute A = [[0]*3]*2 followed by A[0][1] = 7. What is A? Answer: [[0, 7, 0], [0, 7, 0]], because both entries refer to the same row object.

The array sums, deviations, and matrix operations are checked in the Unit III Maxima worksheet; the aliasing result was independently executed in Python, and every displayed Maxima residual is zero.

References

Array (data structure) — Wikipedia.
Python tutorial: Data Structures, Python Software Foundation, sections 5.1–5.6.
Allen B. Downey, Think Python: How to Think Like a Computer Scientist, 2nd ed., Chapter 10, “Lists.”