Step 1 Rule of partitioning of matrix.

Large matrices can be partitioned using a system of horizontal and vertical (dashed) lines.

Clearly a given matrix can be divided into blocks in different ways.

For example, the following \(4 \times 5\) matrix is partitioned into six blocks. \(A=\begin{bmatrix}1 & 2&\vdots &3&7&\vdots&0 \\ 5 & 2& \vdots&9&0&\vdots&2 \\ \dots& \dots&\dots &\dots&\dots&\dots&\dots \\4 & 7& \vdots&0&1&\vdots&4 \\4 & 6&\vdots &7&11&\vdots&2 \end{bmatrix}=\begin{bmatrix}E & F & G \\ H & I & J \end{bmatrix}\)

where E,F,G,H,I and J are the arrays indicated by dashed lines.

The above matrix can also be partitioned as,

\(A=\begin{bmatrix}1 & 2&\vdots &3&\vdots&7&0 \\ \dots& \dots&\dots &\dots&\dots&\dots&\dots\\ 5 & 2& \vdots&9&\vdots&0&2 \\4 & 7& \vdots&0&\vdots&1&4 \\4 & 6&\vdots &7&\vdots&11&2 \end{bmatrix}=\begin{bmatrix}E & F & G \\ H & I & J \end{bmatrix}\)

This shows that a matrix can be partitioned in many ways, but the new blocks can be of different size and different order.

The matrix entries of such a partitioned matrix are called submatrices or blocks. The main matrix is often referred to as super matrix.

Step 2 Importance for partitioning of matrix.

Partitioning of matrices is useful when it is applied to very large matrices because the operations can be carried out easily on the smaller blocks of the matrices. On operating the partitioned matrices, the basic rule can be applied to the blocks as though they were single elements.

If M is a square block matrix such that the non diagonal blocks are all zero matrices, i.e. \(a_ij=0 \text{ when } i \neq j\). then A is called a block diagonal matrix. \(M=diag(A_{11},A_{22},\dots,A_{nn})\)

The importance of block diagonal matrix is that the algebra of the block matrix is frequently reduced to the algebra of the individual blocks. Suppose f(x) is a polynomial and M is the above mentioned block diagonal matrix. Then f(M) is a block diagonal matrix and

\(f(M)=diag(f(A_{11}),f(A_{22}),\dots,f(A_{nn}))\) Also, the matrix M is invertible if and only if each Aii is invertible, and in such case \(M^{-1}\) is a block diagonal matrix, and,

\(M^{-1}=diag(A^{-1}_{11},A^{-1}_{22},\dots,A^{-1}_{nn})\) Also a square block matrix is called a block upper triangular matrix if the blocks below the diagonal are zero matrices, and a block lower triangular matrix if the blocks above the diagonal are zero matrices.