PARTITIONING OF MATRICES
Definition. Some time we partition a matrix into blocks of elements and consider such blocks as elements of the matrix. These blocks are called sub-matrices of the original matrix and are labelled in the same manner as elements of a matrix. A matrix can be partitioned into sub-matrices in many different ways and the manner in which it is to be partitioned is often indicated by drawing horizontal and vertical lines between selected rows and columns.
Earn Money online without investment Earn Money online from home without investment
Let
A=\left[\begin{array}{cccc}a_{11} & a_{12} \therefore & \cdots & a_{1 n} \\ a_{21} & a_{22} & \cdots & a_{2 n} \\ \vdots & \vdots & \cdots & \vdots \\ a_{m 1} & a_{m 2} & \cdots & a_{m n}\end{array}\right]be an [katex]m \times n[/katex] matrix. We can write a partitioning of A as
\begin{aligned}
& A=\left[\begin{array}{ll}B_{11} & B_{12} \\B_{21} & B_{22}\end{array}\right] \text {, where } \\
& B_{11}=\left[\begin{array}{cccc}a_{11} & a_{12} & \cdots & a_{1k} \\a_{21} & a_{22} & \cdots & a_{2k} \\\vdots & \vdots & \cdots & \vdots \\a_{p1} & a_{p2} & \cdots & a_{p k}\end{array}\right] \text {. } \\
& B_{12}=\left[\begin{array}{cccc}a_{1 k+1} & a_{1 k+2} & \cdots & a_{1 n} \\a_{2 k+1} & a_{2 k+2} & \cdots & a_{2 n} \\\vdots & \vdots & \cdots & \vdots \\a_{p k+1} & a_{p k+2} & \cdots & a_{p n}\end{array}\right] \text {. } \\
& B_{21}=\left[\begin{array}{cccc}a_{p+11} & a_{p+12} & \cdots & a_{p+1 k} \\a_{p+21} & a_{p+22} & \cdots & a_{p+2 k} \\\vdots & \vdots & \cdots & \vdots \\a_{m 1} & a_{m 2} & \cdots & a_{m k}\end{array}\right] \\
& B_{22}=\left[\begin{array}{cccc}a_{p+1} k+1 & a_{p+1} k+2 & \cdots & a_{p+1 n} \\a_{p+2 k+1} & a_{p+2 k+2} & \cdots & a_{p+2 n} \\\vdots & \vdots & \cdots & \vdots \\a_{m k+1} & a_{m k+2} & \cdots & a_{m n}\end{array}\right]
\end{aligned}Here [katex]B_{11}[/katex] is a [katex]p \times k[/katex] matrix. [katex]B_{12}[/katex] is a [katex]p \times(n-k)[/katex] matrix. [katex]B_{21}[/katex] is an [katex](m-p) \times k[/katex] matrix. [katex]B_{22}[/katex] is an [katex](m-p) \times(n-k)[/katex] matrix. Let A, B be two [katex]m \times n[/katex] matrices. A and B are said to be identically partitioned if the corresponding submatrices [katex]C_{i j}[/katex] and [katex]D_{i j}[/katex] of A and B are of the same order (i.e. have the same size). In such a case [katex]C_{i j} \pm D_{i j}[/katex] are both defined.
Example. Consider
A=\left[\begin{array}{ccc}
1 & -2 & 5 \\
3 & 0 & 2 \\
4 & 2 & -6
\end{array}\right]A can be partitioned as shown below:
A=\left[\begin{array}{rrr}1 & -2 & 5 \\ 3 & 0 & 2 \\ 4 & 2 & -6\end{array}\right]Example. Partitioning of the matrices A and B are given as under:
A=\left[\begin{array}{rrr}
1 & 2 & 5 \\
-4 & 3 & 0 \\
2 & 1 & 5
\end{array}\right]=\left[\begin{array}{ll}
A_{11} & A_{12} \\
A_{21} & A_{22}
\end{array}\right]and
B=\left[\begin{array}{cc|c}a & b & c \\ d & e & f \\ g & h & t\end{array}\right]=\left[\begin{array}{cc}B_{11} & B_{12} \\ B_{21} & B_{22}\end{array}\right]Here A and B are identically partitioned. [katex]A_{11}[/katex] and [katex]B_{11}[/katex] are [katex]1 \times 2[/katex] submatrices [katex]A_{12}[/katex] and [katex]B_{12}[/katex] are [katex]1 \times 1[/katex] submatrices [katex]A_{21}[/katex] and [katex]B_{21}[/katex] are [katex]2 \times 2[/katex] submatrices [katex]A_{22}[/katex] and [katex]B_{22}[/katex] are [katex]2 \times 1[/katex] submatrices
Moreover, [katex]A_{11}+B_{11}, A_{12}+B_{12}, A_{21}+B_{21}[/katex] and [katex]A_{22}+B_{22}[/katex] are all defined. We have
\begin{aligned}
A+B &=\left[\begin{array}{ll}
A_{11}+B_{11} & A_{12}+B_{12} \\
A_{21}+B_{21} & A_{22}+B_{22}
\end{array}\right] \\
&=\left[\begin{array}{cc|c}
1+a & 2+b & 3+c \\
-4+d & 3+c & 0+f \\
2+g & 1+h & 5+i
\end{array}\right]
\end{aligned}Partitioning can be used in matrix multiplication provided that the matrices are partitioned in such a way that the corresponding submatrices to be multiplied, are conformable for multiplication. Example. Let
A=\left[\begin{array}{ll|l}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33}\end{array}\right]=\left[\begin{array}{ll}A_{11} & A_{12} \\ A_{21} & A_{22}\end{array}\right]\\
B=\left[\begin{array}{ll}b_{11} & b_{12} \\ b_{21} & b_{22} \\ b_{31} & b_{32}\end{array}\right]=\left[\begin{array}{l}B_{11} \\ B_{31}\end{array}\right], where
A_{11}=\left[\begin{array}{ll}a_{11} & a_{12} \\ a_{21} & a_{22}\end{array}\right],\\A_{12}=\left[\begin{array}{ll}a_{13} \\a_{23}\end{array}\right],A_{21}=\left[\begin{array}{ll}a_{31} & a_{32}\end{array}\right],A_{22}=\left[{a}_{33}\right],\\ B_{11}=\left[\begin{array}{ll}b_{11} & b_{12} \\ b_{21} & b_{22}\end{array}\right],\\B_{21}=\left[\begin{array}{ll}b_{31} & b_{32}\end{array}\right],Then
AB=\left[\begin{array}{l}A_{11} B_{11}+A_{12} B_{21} \\ A_{21} B_{11}+A_{22} B_{21}\end{array}\right]\\Now
A_{11}B_{11}=\left[\begin{array}{ll}a_{11} h_{11}+a_{12} b_{21} & a_{11} b_{12}+a_{12} b_{22} \\ a_{21} b_{11}+a_{22} b_{21} & a_{21} b_{12}+a_{22} b_{22}\end{array}\right]A_{12} B_{21}=\left[\begin{array}{ll}a_{13} b_{31} & a_{13} b_{32} \ a_{23} b_{31} & a_{33} b_{23}\end{array}\right],A_{21} B_{11}=\left[\begin{array}{lll}a_{31} b_{11}+a_{32} b_{21} & a_{31} b_{12}+a_{32} b_{22}\end{array}\right]A_{22} B_{21}=\left[a_{33} a_{13}+a_{33} b_{12}\right]Thus
\begin{aligned}
{\left[\begin{array}{l}
A_{11} B_{11}+A_{12} B_{21} \\
A_{21} B_{11}+A_{22} B_{21}
\end{array}\right] } &=\left[\begin{array}{ll}
a_{11} b_{11}+a_{12} b_{21}+a_{13} b_{31} & a_{11} b_{12}+a_{12} b_{22}+a_{13} b_{32} \\
a_{21} b_{11}+a_{22} b_{21}+a_{23} b_{31} & a_{21} b_{12}+a_{22} b_{22}+a_{13} b_{12} \\
a_{31} b_{11}+a_{22} b_{21}+a_{33} b_{31} & a_{31} b_{12}+a_{12} b_{23}+a_{32} b_{31}
\end{array}\right] \\
&=A B .
\end{aligned}This method can be generalized for the product of any pair of matrices which are conformable for multiplication. It is called block multiplication of matrices.