Block Matrix/Examples/Arbitrary 2

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