What Is a Submatrix?

October 11, 2023

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A submatrix of a matrix is one other matrix obtained by forming the intersection of sure rows and columns, or equivalently by deleting sure rows and columns. Extra exactly, let $A$ be an $m\times n$ matrix and let $1\le i_1 < i_2 < \cdots < i_p \le m$ and $1\le j_1 < j_2 < \cdots < j_q \le n$ . Then the $p\times q$ matrix $B$ with $b_{rs} = a_{i_rj_s}$ is the submatrix of $A$ comprising the weather on the intersection of the rows listed by $i_1,\dots,i_p$ and the columns listed by $j_1,\dots,j_q$ . For instance, for the matrix

$\notag \begin{bmatrix} ~\fbox{1} & \fbox{2} & 3\mskip2mu \\ ~~4 & 5 & 6\mskip2mu~ \\ ~\fbox{7} & \fbox{8} & 9~\mskip2mu \end{bmatrix} = \begin{bmatrix} ~\fbox{1} & \fbox{2} & 3\mskip2mu~ \\[1pt] ~\fbox{4} & 5 & 6\mskip2mu~ \\ 7 & \fbox{8} & 9\mskip2mu \end{bmatrix},$

proven with 4 parts highlighted in two alternative ways,

$\notag \begin{bmatrix} 1 & 2 \\ 7 & 8 \\ \end{bmatrix}$

is a submatrix (the intersection of rows $1$ and $3$ and columns $1$ and $2$ , or what’s left after deleting row $2$ and column $3$ ), however

$\notag \begin{bmatrix} 1 & 2 \\ 4 & 8 \\ \end{bmatrix}$

is emph{not} a submatrix.

Submatrices embrace the $mn$ matrix parts and the matrix itself, however there are various of intermediate measurement: an $m\times n$ matrix has $(2^m-1)(2^n-1)$ submatrices in whole (counting each sq. and nonsquare submatrices).

If $p = q$ and $i_k = j_k$ , $k = 1\colon p$ , then $B$ is a principal submatrix of $A$ , which is a submatrix symmetrically situated concerning the diagonal. If, as well as, $i_k = k$ , $k=1\colon p$ , then $B$ is a main principal submatrix of $A$ , which is one located within the prime left nook of $A$ .

The determinant of a sq. submatrix is known as a minor. The Laplace enlargement of the determinant expresses the determinant as a weighted sum of minors.

The Colon Notation

In varied programming languages, notably MATLAB, and in numerical linear algebra, a colon notation is used to indicate submatrices consisting of contiguous rows and columns.

For integers $p$ and $q$ we denote by $p\colon q$ the sequence $p,p+1,\dots,q$ . Thus $i = 1\colon n$ is one other approach of writing $i = 1,2,\dots,n$ .

We write $A(p\colon q,r \colon s)$ for the submatrix of $A$ comprising the intersection of rows $p$ to $q$ and columns $r$ to $s$ , that’s,

$A(p\colon q,r \colon s) = \begin{bmatrix}a_{pr} & \cdots & a_{ps} \\ \vdots & \ddots & \vdots\cr a_{qr} & \cdots & a_{qs} \end{bmatrix}.$

We will consider $A(p\colon q,r \colon s)$ as a projection of $A$ utilizing the corresponding rows and columns of the id matrix:

$\notag A(p\colon q,r \colon s) = I(p\colon q,:) \,A \,I(:,r\colon s).$

As particular circumstances, $A(k,\colon)$ denotes the $k$ th row of $A$ and $A(\colon,k)$ the $k$ th column of $A$ .

Listed below are some examples of utilizing the colon notation to extract submatrices in MATLAB. Rows and columns could be listed by a spread utilizing the colon notation or by specifying the required indices in a vector. The matrix used is from the Anymatrix assortment.

>> A = anymatrix('core/beta',5)
A =
     1     2     3     4     5
     2     6    12    20    30
     3    12    30    60   105
     4    20    60   140   280
     5    30   105   280   630

>> A(3:4, [2 4 5]) % Rectangular submatrix.
ans =
    12    60   105
    20   140   280

>> A(1:2,4:5)      % Sq., however nonprincipal, submatrix.
ans =
     4     5
    20    30

>> A([3 5],[3 5])  % Principal submatrix.
ans =
    30   105
   105   630

Block Matrices

Submatrices are intimately related to block matrices, that are matrices wherein the weather are themselves matrices. For instance, a $4\times 4$ matrix $A$ could be thought to be a block $2\times 2$ matrix, the place every aspect is a $2\times 2$ submatrix of $A$ :

$\notag A = \left[\begin{array}cc a_{11} & a_{12} & a_{13} & a_{14}\\ a_{21} & a_{22} & a_{23} & a_{24}\\\hline a_{31} & a_{32} & a_{33} & a_{34}\\ a_{41} & a_{42} & a_{43} & a_{44}\\ \end{array}\right] = \begin{bmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{bmatrix},$

the place

$\notag A_{11} = A(1\colon2,1\colon2) = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix},$

and likewise for the opposite three blocks.

Associated Weblog Posts

This text is a part of the “What Is” collection, out there from https://nhigham.com/index-of-what-is-articles/ and in PDF kind from the GitHub repository https://github.com/higham/what-is.

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What Is a Submatrix?

The Colon Notation

Block Matrices

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