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.
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
Then, by deleting the second row and the third column of , we obtain the submatrix
By deleting the first column of , we obtain the submatrix
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
Then, by deleting its second row, we get the block
Example Let be the row vector
Then, after striking out its third column, we are left with the submatrix
If we instead delete the first and second column of , we get
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
where an horizontal cut has been performed between the first and the second row. Then, we can write
or simply
where
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,
Thus,
where the four submatrices are
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
Then, we can adjoin the four blocks to create the block matrix
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:
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.
Below you can find some exercises with explained solutions.
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
After performing the cuts, the matrix can be written as
where
Find what partitioned matrix is obtained by horizontally adjoining the blocks
The partitioned matrix is
Please cite as:
Taboga, Marco (2021). "Block matrix", Lectures on matrix algebra. https://www.statlect.com/matrix-algebra/block-matrix.
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