14. Partitioned Matrices
Dated: 18-04-2025
A block matrix or a partitioned matrix is a partition of a matrix1 into rectangular smaller matrices1 called blocks.
Example
\[
P = \begin{bmatrix} 1 & 1 & 2 & 2 \\ 1 & 1 & 2 & 2 \\ 3 & 3 & 4 & 4 \\ 3 & 3 & 4 & 4 \end{bmatrix}
\]
\(P\) can be partitioned as
\[
P_{11} = \begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix}, \quad P_{12} = \begin{bmatrix} 2 & 2 \\ 2 & 2 \end{bmatrix}, \quad P_{21} = \begin{bmatrix} 3 & 3 \\ 3 & 3 \end{bmatrix}, \quad P_{22} = \begin{bmatrix} 4 & 4 \\ 4 & 4 \end{bmatrix}
\]
\[
P = \begin{bmatrix} P_{11} & P_{12} \\ P_{21} & P_{22} \end{bmatrix}
\]
Column Matrices
\[
A = \left[
\begin{array}{c|c|c}
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23} \\
a_{31} & a_{32} & a_{33} \\
a_{41} & a_{42} & a_{43} \\
a_{51} & a_{52} & a_{53} \\
\end{array}
\right]
= \left[ \, C_1 \; \middle| \; C_2 \; \middle| \; C_3 \, \right]
\]
\(C_1, C_2\) and \(C_3\) are called column matrices of \(A\).
Row Matrices
\[
A = \left[
\begin{array}{ccc}
a_{11} & a_{12} & a_{13} \\
\hline
a_{21} & a_{22} & a_{23} \\
\hline
a_{31} & a_{32} & a_{33} \\
\hline
a_{41} & a_{42} & a_{43} \\
\hline
a_{51} & a_{52} & a_{53} \\
\end{array}
\right]
=
\left[
\begin{array}{c}
R_1 \\
\hline
R_2 \\
\hline
R_3 \\
\hline
R_4 \\
\hline
R_5 \\
\end{array}
\right]
\]
\(R_1, R_2, R_3, R_4\) and \(R_5\) are row matrices of \(A\).
Addition of Blocked Matrices
If \(A\) and \(B\) are matrices1 of same size and are partitioned in the same way then each block of \(A + B\) is sum of corresponding blocks of \(A\) and \(B\).
Theorem
If \(A\) is \(m \times n\) and \(B\) is \(n \times p\) then
\[AB =
\begin{bmatrix}
\text{Col}_1(A) & \text{Col}_2(A) & \ldots & \text{Col}_n(A)
\end{bmatrix}
\begin{bmatrix}
\text{Row}_1(B)\\
\text{Row}_2(B)\\
\vdots \\
\text{Row}_n(B)\\
\end{bmatrix}
\]
\[
= \text{Col}_1(A) \text{Row}_1(B) +
\text{Col}_2(A) \text{Row}_2(B) +
\cdots
\text{Col}_n(A) \text{Row}_n(B)
\]
Multiplication of Partitioned Matrices
\[
A = \begin{bmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \\ A_{31} & A_{32} \end{bmatrix} \text{ and } B = \begin{bmatrix} B_{11} & B_{12} \\ B_{21} & B_{22} \end{bmatrix}
\]
\[
AB = \begin{bmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \\ A_{31} & A_{32} \end{bmatrix} \begin{bmatrix} B_{11} & B_{12} \\ B_{21} & B_{22} \end{bmatrix} = \begin{bmatrix} A_{11}B_{11} + A_{12}B_{21} & A_{11}B_{12} + A_{12}B_{22} \\ A_{21}B_{11} + A_{22}B_{21} & A_{21}B_{12} + A_{22}B_{22} \\ A_{31}B_{11} + A_{32}B_{21} & A_{31}B_{12} + A_{32}B_{22} \end{bmatrix}
\]
It is known as block multiplication.
Example
\[
A = \begin{bmatrix} 1 & 0 & 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 & 0 & 1 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 1 & 1 & 1 & 0 & 0 & 1 \end{bmatrix}
\]
Evaluate \(A^2\)
Solution
\[
A = \left[
\begin{array}{ccc|cc|c}
1 & 0 & 0 & 0 & 0 & 1 \\
0 & 1 & 0 & 0 & 0 & 1 \\
0 & 0 & 1 & 0 & 0 & 1 \\
\hline
0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 \\
\hline
1 & 1 & 1 & 0 & 0 & 1
\end{array}
\right]
=
\begin{bmatrix}
I_3 & O_{32} & A_1 \\
O_{23} & I_2 & O_{21} \\
A_1^t & O_{12} & 1
\end{bmatrix}
\]
Where
\[
A_1 =
\begin{bmatrix}
1\\
1\\
1
\end{bmatrix}
\]
\[
A^2 = \begin{bmatrix} I_3 & O_{32} & A_1 \\ O_{23} & I_2 & O_{21} \\ A_1^t & O_{12} & 1 \end{bmatrix} \begin{bmatrix} I_3 & O_{32} & A_1 \\ O_{23} & I_2 & O_{21} \\ A_1^t & O_{12} & 1 \end{bmatrix} = \begin{bmatrix} I_3 + A_1A_1^t & O_{32} & A_1 + A_1 \\ O_{23} & I_2 & O_{21} \\ A_1^t + A_1^t & O_{12} & A_1^t A_1 + 1 \end{bmatrix}
\]
\[
I_3 + A_1A_1^t = \begin{bmatrix} 2 & 1 & 1 \\ 1 & 2 & 1 \\ 1 & 1 & 2 \end{bmatrix}
\]
\[
A_1 + A_1 = \begin{bmatrix} 2 \\ 2 \\ 2 \end{bmatrix}
\]
\[
A_1^t + A_1^t = \begin{bmatrix} 2 & 2 & 2 \end{bmatrix}
\]
\[
A_1^t A_1 + 1 = [4]
\]
\[
A^2 =
\left[
\begin{array}{ccc|cc|c}
2 & 1 & 1 & 0 & 0 & 2 \\
1 & 2 & 1 & 0 & 0 & 2 \\
1 & 1 & 2 & 0 & 0 & 2 \\
\hline
0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 \\
\hline
2 & 2 & 2 & 0 & 0 & 4
\end{array}
\right]
\]
\[
A^2 =
\begin{bmatrix}
2 & 1 & 1 & 0 & 0 & 2 \\
1 & 2 & 1 & 0 & 0 & 2 \\
1 & 1 & 2 & 0 & 0 & 2 \\
0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 \\
2 & 2 & 2 & 0 & 0 & 4
\end{bmatrix}
\]
Toeplitz Matrix
A matrix1 in which each descending diagonal from left to right is constant is called a Toeplitz matrix or diagonal-constant matrix.
Example
\[
A = \begin{bmatrix} a & 1 & 2 & 3 \\ 4 & a & 1 & 2 \\ 5 & 4 & a & 1 \\ 6 & 5 & 4 & a \end{bmatrix}
\]
Block Toeplitz Matrix
A blocked matrix in which blocks (blocked matrices) are repeated down the diagonals of the matrix1 is called a blocked Toeplitz matrix.
\[
B = \begin{bmatrix}
B(1,1) & B(1,2) & B(1,3) & B(1,4) & B(1,5) \\
B(2,1) & B(1,1) & B(1,2) & B(1,3) & B(1,4) \\
B(3,1) & B(2,1) & B(1,1) & B(1,2) & B(1,3) \\
B(4,1) & B(3,1) & B(2,1) & B(1,1) & B(1,2) \\
B(5,1) & B(4,1) & B(3,1) & B(2,1) & B(1,1)
\end{bmatrix}
\]
Inverses of Partitioned Matrices
Block Diagonal Matrices
\[
A = \begin{bmatrix}
D_1 & 0 & \cdots & 0 \\
0 & D_2 & \cdots & 0 \\
\vdots & \vdots & \ddots & \vdots \\
0 & 0 & \cdots & D_k
\end{bmatrix}
\]
Where \(D_1, D_2\) and \(D_k\) are square matrices.1
It's inverse can be found out as
\[
A^{-1} = \begin{bmatrix}
D_1^{-1} & 0 & \cdots & 0 \\
0 & D_2^{-1} & \cdots & 0 \\
\vdots & \vdots & \ddots & \vdots \\
0 & 0 & \cdots & D_k^{-1}
\end{bmatrix}
\]
Block Upper Triangular Matrices
\[
A = \begin{bmatrix}
A_{11} & A_{12} & \cdots & A_{1k} \\
O & A_{22} & \cdots & A_{2k} \\
\vdots & \vdots & \ddots & \vdots \\
O & O & \cdots & A_{kk}
\end{bmatrix}
\]
Where \(A_{11}, A_{22}\) and \(A_{kk}\) are square.
Example
Let \(A\) be a block upper triangular matrix.
\[
A =
\begin{bmatrix}
A_{11} & A_{12}\\
0 & A_{22}
\end{bmatrix}
\]
Where the order1 of \(A_{11}\) is \(p \times p\) and that of \(A_{22}\) is \(q \times q\) respectively.
Find \(A^{-1}\)
Solution
\[
B =
\begin{bmatrix}
B_{11} & B_{12}\\
B_{21} & B_{22}
\end{bmatrix}
\]
\[B = A^{-1}\]
\[
AB = \begin{bmatrix} A_{11} & A_{12} \\ O & A_{22} \end{bmatrix} \begin{bmatrix} B_{11} & B_{12} \\ B_{21} & B_{22} \end{bmatrix} = \begin{bmatrix} I_p & O \\ O & I_q \end{bmatrix}
\]
\[
\begin{bmatrix} A_{11}B_{11} + A_{12}B_{21} & A_{11}B_{12} + A_{12}B_{22} \\ A_{22}B_{21} & A_{22}B_{22} \end{bmatrix} = \begin{bmatrix} I_p & O \\ O & I_q \end{bmatrix}
\]
\[
A_{11}B_{11} + A_{12}B_{21} = I_p
\]
\[
A_{11}B_{12} + A_{12}B_{22} = O
\]
\[
A_{22}B_{21} = O
\]
\[
A_{22}B_{22} = I_q
\]
\(B_{22} = ?\)
\[
A_{22}B_{22} = I_q
\]
\[
A_{22}^{-1}A_{22}B_{22} = A^{-1}_{22}I_q
\]
\[
I_{q}B_{22} = A^{-1}_{22}
\]
\[
B_{22} = A^{-1}_{22}
\]
\(B_{21} = ?\)
\[
A_{22}B_{21} = O
\]
\[
A_{22}^{-1}A_{22}B_{21} = A^{-1}_{22}O
\]
\[
I_{q}B_{21} = O
\]
\[
B_{21} = O
\]
\(B_{11} = ?\)
\[
A_{11}B_{11} + O = I_p \\
\implies A_{11}B_{11} = I_p \\
\implies B_{11} = A_{11}^{-1}
\]
\(B_{12} = ?\)
\[
A_{11}B_{12} + A_{12}B_{22} = O
\]
\[
A_{11}B_{12} = -A_{12}B_{22}
\]
\[
A^{-1}_{11}A_{11}B_{12} = A_{11}^{-1}-A_{12}B_{22}
\]
\[
I_1B_{12} = A_{11}^{-1}-A_{12}B_{22}
\]
\[
B_{12} = A_{11}^{-1}-A_{12}A_{22}^{-1} \quad \because B_{22}=A^{-1}_{22}
\]
\[
A^{-1} = \begin{bmatrix} A_{11} & A_{12} \\ O & A_{22} \end{bmatrix}^{-1} = \begin{bmatrix} A_{11}^{-1} & -A_{11}^{-1}A_{12}A_{22}^{-1} \\ O & A_{22}^{-1} \end{bmatrix}
\]
References
Read more about notations and symbols.