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

$P={\begin{bmatrix}1&1&2&2\\1&1&2&2\\3&3&4&4\\3&3&4&4\end{bmatrix}}$

$P_{11}={\begin{bmatrix}1&1\\1&1\end{bmatrix}},P_{12}={\begin{bmatrix}2&2\\2&2\end{bmatrix}},P_{21}={\begin{bmatrix}3&3\\3&3\end{bmatrix}},P_{22}={\begin{bmatrix}4&4\\4&4\end{bmatrix}}$

Then we can write the partitioned matrix like this

$P_{\text{partitioned}}={\begin{bmatrix}P_{11}&P_{12}\\P_{21}&P_{22}\end{bmatrix}}$