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Block matrix

by , PhD

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

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

Example Define [eq1]Then, by deleting the second row and the third column of $M$, we obtain the submatrix[eq2]By deleting the first column of $M$, 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 $M$ 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 $M$, we get[eq8]

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 $M$ is made up of the two blocks A and $B$.

Example Take the block matrix $M$ 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:

Example Consider the following matrix with six blocks:[eq18]Then, for instance, A, $B$ and $C$ must have the same number of rows and $B$ and E 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 A, $B$ and $C$ are adjoined horizontally, while $B$ and E 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 (2017). "Block matrix", Lectures on matrix algebra. https://www.statlect.com/matrix-algebra/block-matrix.

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