# 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.

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

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

If A is invertible, its inverse is

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

where

 $\displaystyle\mathbf{{B}_{11}}$ $\displaystyle=\mathbf{{({A}_{11}-{A}_{12}{A}_{22}^{-1}{A}_{21})}^{-1}}$ $\displaystyle\mathbf{{B}_{12}}$ $\displaystyle=\mathbf{-{A}_{11}^{-1}{A}_{12}{B}_{22}}$ (32) $\displaystyle\mathbf{{B}_{21}}$ $\displaystyle=\mathbf{-{A}_{22}^{-1}{A}_{21}{B}_{11}}$ $\displaystyle\mathbf{{B}_{22}}$ $\displaystyle=\mathbf{{({A}_{22}-{A}_{21}{A}_{11}^{-1}{A}_{12})}^{-1}}$

Alternately,

 $\displaystyle\mathbf{{B}_{12}}$ $\displaystyle=\mathbf{-{B}_{11}{A}_{12}{A}_{22}^{-1}}$ (33) $\displaystyle\mathbf{{B}_{22}}$ $\displaystyle=\mathbf{{A}_{22}^{-1}-{A}_{22}^{-1}{A}_{21}{B}_{12}}$

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

 $\displaystyle\mathbf{C_{11}}$ $\displaystyle=\mathbf{A_{11}B_{11}+A_{12}B_{21}}$ $\displaystyle\mathbf{C_{12}}$ $\displaystyle=\mathbf{A_{11}B_{12}+A_{12}B_{22}}$ (34) $\displaystyle\mathbf{C_{21}}$ $\displaystyle=\mathbf{A_{21}B_{11}+A_{22}B_{21}}$ $\displaystyle\mathbf{C_{22}}$ $\displaystyle=\mathbf{A_{21}B_{12}+A_{22}B_{22}}$

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.