Null Matrix

TL;DR

• Every entry of a null matrix is 0.
• Adding a null matrix to any other matrix does not change the other matrix’s values.
• Null matrices can have any size or shape and are often used as placeholders in programming (e.g., in Python as an empty list or a list of zeros).

Definition

A null matrix, also known as a zero matrix, is a matrix in which every element is equal to 0.

Explanation

A null matrix contains only zeros in every position. It appears across mathematics and computer science because of its behavior in matrix operations: in matrix addition, a null matrix can be added to any other matrix without changing the other matrix’s values. In programming contexts, null matrices are commonly used as placeholders for matrices that are not yet defined; for example, in languages like Python a null matrix may be represented as an empty list or a list of zeros that can later be filled.

Null matrices may be of any dimension (any number of rows and columns) provided all entries are zero.

Examples

3×3 null matrix

$$\begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}$$

1×1 null matrix

$$\begin{bmatrix} 0 \end{bmatrix}$$

5×5 null matrix

$$\begin{bmatrix} 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{bmatrix}$$

2×3 null matrix

$$\begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}$$

Use cases

• In mathematics, particularly linear algebra, null matrices are used in matrix operations because of their special properties.
• In computer science, null matrices serve as placeholders for matrices that are not yet defined; in programming languages like Python they can be represented as an empty list or a list of zeros to be filled later.

Related terms

• Zero matrix
• Matrix addition
• Linear algebra
• Python