Block Matrix/Examples/Arbitrary 2
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Example of Block Matrix
Consider the matrices:
- $\mathbf A := \sqbrk \alpha_{m n}$: an $m \times n$ matrix
- $\mathbf B := \sqbrk \beta_{p m}$: a $p \times m$ matrix
over an arbitrary ring $R$.
Then:
- $\mathbf M = \begin{bmatrix}
\mathbf I_m & \mathbf A \\ \mathbf B & \mathbf 0 \end{bmatrix}$
is a matrix whose order is $\paren {m + p} \times \paren {m + n}$ where:
- $\mathbf I_m$ is the unit matrix of order $m$
- $\mathbf 0$ is the zero matrix of order $p \times n$.
$\mathbf M := \sqbrk \gamma_{\paren {m + p}, \paren {m + n} }$ can be represented as:
- $\gamma_{i j} = \begin {cases}
\delta_{i j} & : \tuple {i, j} \in \closedint 1 m \times \closedint 1 m \\ \alpha_{i, j - m} & : \tuple {i, j} \in \closedint 1 m \times \closedint {m + 1} {m + n} \\ \beta_{i - m, j} & : \tuple {i, j} \in \closedint {m + 1} {m + p} \times \closedint 1 m \\ 0_R & : \tuple {i, j} \in \closedint {m + 1} {m + p} \times \closedint {m + 1} {m + n} \end {cases}$
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 29$. Matrices