Block matrix

A block matrix (or partitioned matrix) is a matrix that is subdivided into blocks that are themselves matrices. The subdivision is performed by cutting the matrix one or more times, vertically and/or horizontally.

Table of contents

Blocks
Horizontal and vertical cuts
Adjoining blocks
Solved exercises
1. Exercise 1
2. Exercise 2

Blocks

Given a matrix , a submatrix (or block) of is a matrix that is obtained from by deleting some of its rows and/or columns.

Example Define [eq1] Then, by deleting the second row and the third column of , we obtain the submatrix [eq2] By deleting the first column of , we obtain the submatrix [eq3]

Row and column vectors, despite being special matrices that have a single column or row respectively, can be used to form blocks.

Example Consider the column vector [eq4] Then, by deleting its second row, we get the block [eq5]

Example Let be the row vector [eq6] Then, after striking out its third column, we are left with the submatrix [eq7] If we instead delete the first and second column of , we get

Horizontal and vertical cuts

As we said in the introduction, a block matrix is the result of performing some vertical and horizontal cuts on a matrix so as to subdivide it into blocks.

Example Define [eq9] where an horizontal cut has been performed between the first and the second row. Then, we can write [eq10] or simply [eq11] where [eq12] Thus, the partitioned matrix is made up of the two blocks and .

Example Take the block matrix in the previous example and perform another cut, vertically, between the first and the second column. Then, [eq13] Thus, [eq14] where the four submatrices are [eq15]

Adjoining blocks

We have seen how to obtain a partitioned matrix by cutting it into blocks. Another way to obtain a partitioned matrix is to first specify the blocks and then adjoin them so as to obtain a larger matrix.

Example Define [eq16] Then, we can adjoin the four blocks to create the block matrix [eq17]

As the cuts between rows and columns cannot be staggered, we need to follow these rules:

if two or more matrices are adjoined on a row, then they must have the same number of rows;
if two or more matrices are adjoined on a column, then they must have the same number of columns.

Example Consider the following matrix with six blocks: [eq18] Then, for instance, , and must have the same number of rows and and must have the same number of columns.

When matrices are adjoined on a row, we say that they are adjoined horizontally. When they are adjoined on a column, we say that they are adjoined vertically.

Example In the previous example , and are adjoined horizontally, while and are adjoined vertically.

Solved exercises

Below you can find some exercises with explained solutions.

Exercise 1

Explicitly write out the blocks that result from performing 1) a horizontal cut between the first and second row, and 2) a vertical cut between the second and third column of the matrix [eq19]

Solution

After performing the cuts, the matrix can be written as [eq14] where [eq21]

Exercise 2

Find what partitioned matrix is obtained by horizontally adjoining the blocks [eq22]

Solution

The partitioned matrix is [eq23]

How to cite

Please cite as:

Taboga, Marco (2021). "Block matrix", Lectures on matrix algebra. https://www.statlect.com/matrix-algebra/block-matrix.