2.9 Partitioning a Matrix

Matrices may be divided into subsections for computational purposes. Consider the n × n matrix A which is partitioned along the following lines.

𝐀=(𝐀𝟏𝟏𝐀𝟏𝟐𝐀𝟐𝟏𝐀𝟐𝟐)\mathbf{A}=\left(\begin{array}[]{cc}\mathbf{A_{11}}&\mathbf{A_{12}}\\ \mathbf{A_{21}}&\mathbf{A_{22}}\end{array}\right) (29)

If k × k matrix A11 and p × p matrix A22 are square matrices, then A12 has dimensions k × p and A21 has dimensions p × k.

The transpose of A is

𝐀𝐓=(𝐀𝟏𝟏𝐓𝐀𝟏𝟐𝐓𝐀𝟐𝟏𝐓𝐀𝟐𝟐𝐓)\mathbf{{A}^{T}}=\left(\begin{array}[]{cc}\mathbf{{A}_{11}^{T}}&\mathbf{{A}_{1% 2}^{T}}\\ \mathbf{{A}_{21}^{T}}&\mathbf{{A}_{22}^{T}}\end{array}\right) (30)

If A is invertible, its inverse is

𝐀-𝟏=(𝐁𝟏𝟏𝐁𝟏𝟐𝐁𝟐𝟏𝐁𝟐𝟐)\mathbf{{A}^{-1}}=\left(\begin{array}[]{cc}\mathbf{{B}_{11}}&\mathbf{{B}_{12}}% \\ \mathbf{{B}_{21}}&\mathbf{{B}_{22}}\end{array}\right) (31)

where

𝐁𝟏𝟏\displaystyle\mathbf{{B}_{11}} =(𝐀𝟏𝟏-𝐀𝟏𝟐𝐀𝟐𝟐-𝟏𝐀𝟐𝟏)-𝟏\displaystyle=\mathbf{{({A}_{11}-{A}_{12}{A}_{22}^{-1}{A}_{21})}^{-1}}
𝐁𝟏𝟐\displaystyle\mathbf{{B}_{12}} =-𝐀𝟏𝟏-𝟏𝐀𝟏𝟐𝐁𝟐𝟐\displaystyle=\mathbf{-{A}_{11}^{-1}{A}_{12}{B}_{22}} (32)
𝐁𝟐𝟏\displaystyle\mathbf{{B}_{21}} =-𝐀𝟐𝟐-𝟏𝐀𝟐𝟏𝐁𝟏𝟏\displaystyle=\mathbf{-{A}_{22}^{-1}{A}_{21}{B}_{11}}
𝐁𝟐𝟐\displaystyle\mathbf{{B}_{22}} =(𝐀𝟐𝟐-𝐀𝟐𝟏𝐀𝟏𝟏-𝟏𝐀𝟏𝟐)-𝟏\displaystyle=\mathbf{{({A}_{22}-{A}_{21}{A}_{11}^{-1}{A}_{12})}^{-1}}

Alternately,

𝐁𝟏𝟐\displaystyle\mathbf{{B}_{12}} =-𝐁𝟏𝟏𝐀𝟏𝟐𝐀𝟐𝟐-𝟏\displaystyle=\mathbf{-{B}_{11}{A}_{12}{A}_{22}^{-1}} (33)
𝐁𝟐𝟐\displaystyle\mathbf{{B}_{22}} =𝐀𝟐𝟐-𝟏-𝐀𝟐𝟐-𝟏𝐀𝟐𝟏𝐁𝟏𝟐\displaystyle=\mathbf{{A}_{22}^{-1}-{A}_{22}^{-1}{A}_{21}{B}_{12}}

The product of A and another n × n matrix B which is partitioned along the the same lines is an identically partitioned matrix C such that

𝐂𝟏𝟏\displaystyle\mathbf{C_{11}} =𝐀𝟏𝟏𝐁𝟏𝟏+𝐀𝟏𝟐𝐁𝟐𝟏\displaystyle=\mathbf{A_{11}B_{11}+A_{12}B_{21}}
𝐂𝟏𝟐\displaystyle\mathbf{C_{12}} =𝐀𝟏𝟏𝐁𝟏𝟐+𝐀𝟏𝟐𝐁𝟐𝟐\displaystyle=\mathbf{A_{11}B_{12}+A_{12}B_{22}} (34)
𝐂𝟐𝟏\displaystyle\mathbf{C_{21}} =𝐀𝟐𝟏𝐁𝟏𝟏+𝐀𝟐𝟐𝐁𝟐𝟏\displaystyle=\mathbf{A_{21}B_{11}+A_{22}B_{21}}
𝐂𝟐𝟐\displaystyle\mathbf{C_{22}} =𝐀𝟐𝟏𝐁𝟏𝟐+𝐀𝟐𝟐𝐁𝟐𝟐\displaystyle=\mathbf{A_{21}B_{12}+A_{22}B_{22}}

The current discussion has focused on the principal partition of a square matrix; however, all aspects of the discussion (except the inversion rules) are more general – provided the dimensions of the partitions are conformable for the indicated operations.