Linear Algebra/Partitioned Matrices

A partitioned matrix also called a block matrix is a partition of a matrix into rectangular smaller matrices called blocks.

The matrix $P$ can be partitioned into 4 2×2 blocks

$P=\left[\begin{array}{cccc}1& 1& 2& 2\\ 1& 1& 2& 2\\ 3& 3& 4& 4\\ 3& 3& 4& 4\end{array}\right]$

$P_{11}=\left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right],P_{12}=\left[\begin{array}{cc}2& 2\\ 2& 2\end{array}\right],P_{21}=\left[\begin{array}{cc}3& 3\\ 3& 3\end{array}\right],P_{22}=\left[\begin{array}{cc}4& 4\\ 4& 4\end{array}\right]$

Then we can write the partitioned matrix like this

$P_{\text{partitioned}}=\left[\begin{array}{cc}{P}_{11}& P_{12}\\ P_{21}& P_{22}\end{array}\right]$