Partitioned Matrices and Submatrix

Definition

Partitioned matrix is a matrix obtained from dividing a bigger matrix by inserting horizontal and vertical rules between selected rows and columns.

Submatrix is a matrix obtained by deleting its rows or columns. A matrix is also considered to be a submatrix of itself.

Remark

The difference is:

• Submatrix subtracts elements. Usually for analysis ( determinants, ranks)
• Partitioning divides a matrix into multiple submatrices. Usually for computations (block operations)

Example

For example, the following are three possible partitions of a general $3\times 4$ matrix $A$:

Partition of $A$ into four submatrices $A_{11},A_{12},A_{21}$ and $A_{22}$ :

$$A=\left[\begin{array}{cccc}{a}_{11}& a_{12}& a_{13}& a_{14}\\ a_{22}& a_{22}& a_{23}& a_{24}\\ a_{11}& a_{12}& a_{12}& a_{14}\end{array}\right]=\left[\begin{array}{cc}{A}_{11}& A_{12}\\ A_{21}& A_{22}\end{array}\right]$$

Partition of $A$ into row vectors ${\mathbf{r}}_1,{\mathbf{r}}_2$ and ${\mathbf{r}}_3$ :

A=\left[ \begin{array} a_{11} & a_{12} & a_{13} & a_{14} \\ \hline a_{22} & a_{22} & a_{23} & a_{24} \\ \hline a_{11} & a_{12} & a_{12} & a_{14} \\ \end{array} \right] =\begin{bmatrix} \mathbf{r}_{1} \\ \mathbf{r}_{2} \\ \mathbf{r}_{3} \\ \end{bmatrix}

Partition of $A$ into column vectors ${\mathbf{c}}_1,{\mathbf{c}}_2$ and ${\mathbf{c}}_3$ :

$$A=\left[\begin{array}{cccc}{a}_{11}& a_{12}& a_{13}& a_{14}\\ a_{22}& a_{22}& a_{23}& a_{24}\\ a_{11}& a_{12}& a_{12}& a_{14}\end{array}\right]=\left[\begin{array}{cccc}{\mathbf{c}}_1& {\mathbf{c}}_2& {\mathbf{c}}_3& {\mathbf{c}}_4\end{array}\right]$$