Definition:Submatrix

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Definition

Let $\mathbf A$ be a matrix with $m$ rows and $n$ columns.


A submatrix of $\mathbf A$ is a matrix formed by selecting from $\mathbf A$:

a subset of the rows

and:

a subset of the columns

and forming a new matrix by using those entries, in the same relative positions, that appear in both the rows and columns of those selected.


Notation

A submatrix of $\mathbf A$ is denoted as follows.

Let:

$\left\{ {a_1, a_2, \ldots, a_r}\right\}$ be the indices of the $r$ selected rows
$\left\{ {b_1, b_2, \ldots, b_s}\right\}$ be the indices of the $s$ selected columns

where all of $a_1, \ldots, a_r$ are between $1$ and $m$, and all of $b_1, \ldots, b_s$ are between $1$ and $n$.


Then the submatrix formed from rows $\left\{ {a_1, a_2, \ldots, a_r}\right\}$ and columns $\left\{ {b_1, b_2, \ldots, b_s}\right\}$ is denoted as:

$\mathbf A \left[{a_1, a_2, \ldots, a_r; b_1, b_2, \ldots, b_s}\right]$

It is usual to specify the rows and columns in ascending numerical order.


Submatrix of order $\left({m - 1}\right) \times \left({n - 1}\right)$

Let a submatrix $\mathbf B$ of $\mathbf A$ be of order $\left({m - 1}\right) \times \left({n - 1}\right)$.

Then it is usual to denote $\mathbf B$ by indicating the (single) row and column of $\mathbf A$ which has been removed, as follows:


Let:

$a_j$ be the row of $\mathbf A$ which is not included in $\mathbf B$
$b_k$ be the column of $\mathbf A$ which is not included in $\mathbf B$.


Then the submatrix $\mathbf B$ formed from the remaining rows and columns of $\mathbf A$ can be denoted as:

$\mathbf A \left({a_j; b_k}\right)$


Example

Let $\mathbf A$ be the $3 \times 4$ matrix defined as follows:

$\mathbf A = \begin{bmatrix} a_{11} & a_{12} & a_{13} & a_{14} \\ a_{21} & a_{22} & a_{23} & a_{24} \\ a_{31} & a_{32} & a_{33} & a_{34} \end{bmatrix}$

Then:

$\mathbf A \left[{1, 2; 1, 3, 4}\right] = \begin{bmatrix} a_{11} & a_{13} & a_{14} \\ a_{21} & a_{23} & a_{24} \end{bmatrix}$

is a submatrix of $\mathbf A$ formed by rows $1, 2$ and columns $1, 3, 4$.


This submatrix can also be denoted by $\mathbf A \left({3; 2}\right)$ which means that it is formed by deleting row $3$ and column $2$.


Also known as

A submatrix can also be called a segment of a matrix.


Also see

The equivalent term for a determinant is a minor.