The "dimension" of a null matrix can be interpreted in two primary ways: referring to its order (its size, i.e., number of rows by number of columns) or, less commonly but relevant in certain mathematical contexts, the dimension of its null space (also known as its nullity). Both interpretations provide a complete understanding of a null matrix's characteristics.
Understanding "Dimension" in Matrix Contexts
In linear algebra, the term "dimension" can apply to different aspects of a matrix:
- Matrix Order: This is the most common and intuitive meaning, describing the matrix's shape as $m \times n$, where 'm' is the number of rows and 'n' is the number of columns.
- Nullity: This refers to the dimension of the null space of a matrix. The null space consists of all vectors that, when multiplied by the matrix, result in the zero vector. Its dimension (the nullity) indicates the number of linearly independent vectors that get mapped to zero by the transformation represented by the matrix.
Interpretation 1: The Order (Size) of a Null Matrix
A null matrix (also known as a zero matrix) is a matrix in which every element is zero. It is denoted by $\mathbf{0}$ or $\mathbf{O}_{m \times n}$ to specify its dimensions.
The dimension of a null matrix, in this context, is simply its size, expressed as its number of rows by its number of columns.
Examples:
- A $2 \times 2$ null matrix has a dimension of $2 \times 2$:
$$ \begin{bmatrix} 0 & 0 \ 0 & 0 \end{bmatrix} $$ - A $3 \times 1$ null matrix (a zero column vector) has a dimension of $3 \times 1$:
$$ \begin{bmatrix} 0 \ 0 \ 0 \end{bmatrix} $$
Dimension Examples for Null Matrices
| Type of Null Matrix | Dimension (Rows $\times$ Columns) | Example Matrix |
|---|---|---|
| $2 \times 2$ Null Matrix | $2 \times 2$ | $\begin{bmatrix} 0 & 0 \ 0 & 0 \end{bmatrix}$ |
| $1 \times 3$ Null Matrix | $1 \times 3$ | $\begin{bmatrix} 0 & 0 & 0 \end{bmatrix}$ |
| $3 \times 2$ Null Matrix | $3 \times 2$ | $\begin{bmatrix} 0 & 0 \ 0 & 0 \ 0 & 0 \end{bmatrix}$ |
Interpretation 2: The Nullity (Dimension of the Null Space) of a Null Matrix
When considering the "dimension of a null matrix" from the perspective of its null space, we are talking about its nullity. As stated in various mathematical resources, the dimension of the null space of a matrix is called the nullity of that matrix.
To calculate the nullity of a matrix, we use the Rank-Nullity Theorem, which provides the following relationship:
$$\text{Nullity of a matrix A} = \text{Number of Columns of A} - \text{Rank of A}$$
Let's apply this to an $m \times n$ null matrix, denoted as $\mathbf{0}_{m \times n}$:
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Rank of a Null Matrix: The rank of a matrix is the maximum number of linearly independent column vectors (or row vectors) in the matrix. For any null matrix, all its rows and columns consist entirely of zeros. Therefore, there are no non-zero linearly independent rows or columns. This means the rank of any null matrix is always 0.
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Calculating Nullity for a Null Matrix:
- Consider an $m \times n$ null matrix $\mathbf{0}_{m \times n}$.
- Number of Columns of $\mathbf{0}_{m \times n}$ is $n$.
- Rank of $\mathbf{0}_{m \times n}$ is $0$.
Using the formula:
$$\text{Nullity of } \mathbf{0}_{m \times n} = n - 0 = n$$
Therefore, the dimension of the null space (nullity) of an $m \times n$ null matrix is equal to its number of columns, $n$.
Example:
Consider a $2 \times 3$ null matrix:
$$ \mathbf{0}_{2 \times 3} = \begin{bmatrix} 0 & 0 & 0 \ 0 & 0 & 0 \end{bmatrix} $$
- Number of columns ($n$) = 3
- Rank of $\mathbf{0}_{2 \times 3}$ = 0
- Nullity of $\mathbf{0}_{2 \times 3}$ = $3 - 0 = 3$
This means that the null space of a $2 \times 3$ null matrix is a 3-dimensional space ($\mathbb{R}^3$), as any vector in $\mathbb{R}^3$ when multiplied by this zero matrix will result in the zero vector.
In summary, while the "dimension of a null matrix" most commonly refers to its order ($m \times n$), in the context of vector spaces and transformations, its nullity (the dimension of its null space) is a crucial characteristic, always equal to its number of columns.
