Definitions

Matrix addition

Matrix addition

In mathematics, matrix addition is the operation of adding two matrices by adding the corresponding entries together. However, there is another operation which could also be considered as a kind of addition for matrices.

Entrywise sum

The usual matrix addition is defined for two matrices of the same dimensions. The sum of two m-by-n matrices A and B, denoted by A + B, is again an m-by-n matrix computed by adding corresponding elements. For example:

begin{bmatrix}
   1 & 3 
   1 & 0 
   1 & 2
end{bmatrix} + begin{bmatrix}
   0 & 0 
   7 & 5 
   2 & 1
end{bmatrix} = begin{bmatrix}
   1+0 & 3+0 
   1+7 & 0+5 
   1+2 & 2+1
end{bmatrix} = begin{bmatrix}
   1 & 3 
   8 & 5 
   3 & 3
end{bmatrix}

We can also subtract one matrix from another, as long as they have the same dimensions. A - B is computed by subtracting corresponding elements of A and B, and has the same dimensions as A and B. For example:

begin{bmatrix}
   1 & 3 
   1 & 0     1 & 2
end{bmatrix} - begin{bmatrix}
   0 & 0 
   7 & 5 
   2 & 1
end{bmatrix} = begin{bmatrix}
   1-0 & 3-0 
   1-7 & 0-5 
   1-2 & 2-1
end{bmatrix} = begin{bmatrix}
   1 & 3 
   -6 & -5 
   -1 & 1
end{bmatrix}

Direct sum

Another operation, which is used less often, is the direct sum. We can form the direct sum of any pair of matrices A and B. say of size m × n and p × q, respectively. The direct sum is a matrix of size (m + p) × (n + q) matrix defined as

 A oplus B =
begin{bmatrix} A & 0 0 & B end{bmatrix} = begin{bmatrix} a_{11} & cdots & a_{1n} & 0 & cdots & 0
    vdots & cdots & vdots & vdots & cdots & vdots 
a_{m 1} & cdots & a_{mn} & 0 & cdots & 0 0 & cdots & 0 & b_{11} & cdots & b_{1q}
    vdots & cdots & vdots & vdots & cdots & 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}

Note that the direct sum of two square matrices could represent the adjacency matrix of a graph or multigraph with one component for each direct addend.

Note also that any element in the direct sum of two vector spaces of matrices could be represented as a direct sum of two matrices.

In general, we can write the direct sum of n matrices as:

bigoplus_{i=1}^{n} A_{i} = mbox{diag}(A_1, A_2, A_3, ldots, A_n)= begin{bmatrix}
     A_1  &  &  &   
     & A_2  &   &   
     &   & ddots  &   
     &   &   & A_n
end{bmatrix}.

See also

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