Matrix Addition - Direct Sum

Direct Sum

Another operation, which is used less often, is the direct sum (denoted by ⊕). Note the Kronecker sum is also denoted ⊕; the context should make the usage clear. The direct sum of any pair of matrices A of size m × n and B of size p × q is a matrix of size (m + p) × (n + q) defined as

$\bold{A} \oplus \bold{B} = \begin{bmatrix} \bold{A} & \boldsymbol{0} \\ \boldsymbol{0} & \bold{B} \end{bmatrix} = \begin{bmatrix} a_{11} & \cdots & a_{1n} & 0 & \cdots & 0 \\ \vdots & \ddots & \vdots & \vdots & \ddots & \vdots \\ a_{m 1} & \cdots & a_{mn} & 0 & \cdots & 0 \\ 0 & \cdots & 0 & b_{11} & \cdots & b_{1q} \\ \vdots & \ddots & \vdots & \vdots & \ddots & \vdots \\ 0 & \cdots & 0 & b_{p1} & \cdots & b_{pq} \end{bmatrix}$

For instance,

$\begin{bmatrix} 1 & 3 & 2 \\ 2 & 3 & 1 \end{bmatrix} \oplus \begin{bmatrix} 1 & 6 \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} 1 & 3 & 2 & 0 & 0 \\ 2 & 3 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 6 \\ 0 & 0 & 0 & 0 & 1 \end{bmatrix}$

The direct sum of matrices is a special type of block matrix, in particular the direct sum of square matrices is a block diagonal matrix.

The adjacency matrix of the union of disjoint graphs or multigraphs is the direct sum of their adjacency matrices. Any element in the direct sum of two vector spaces of matrices can be represented as a direct sum of two matrices.

In general, the direct sum of n matrices is:

$\bigoplus_{i=1}^{n} \bold{A}_{i} = {\rm diag}( \bold{A}_1, \bold{A}_2, \bold{A}_3 \cdots \bold{A}_n)= \begin{bmatrix} \bold{A}_1 & \boldsymbol{0} & \cdots & \boldsymbol{0} \\ \boldsymbol{0} & \bold{A}_2 & \cdots & \boldsymbol{0} \\ \vdots & \vdots & \ddots & \vdots \\ \boldsymbol{0} & \boldsymbol{0} & \cdots & \bold{A}_n \\ \end{bmatrix}\,\!$

where the zeros are actually blocks of zeros, i.e. zero matricies.

NB: Sometimes in this context, boldtype for matrices is dropped, matricies are written in italic.

Read more about this topic: Matrix Addition

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