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

Entrywise sum

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The usual matrix addition is defined for two matrices of 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}

Direct sum

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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 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}.

Alternative matrix addition

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Some statisticians use an alternative matrix addition, which is defined analogous to matrix multiplication. It can be used for concise expression of several key theorems of statistical theory and theory of probability. To add two matrices A and B and store the results in a matrix C

$C\; =\; A\; +\; B$

the number of columns in matrix A must equal the number of rows in matrix B, in other words, the matrices must be conformable to matrix addition. The resulting matrix C will have the number of rows of the first matrix and the number of columns of the second matrix. For example, if matrix A is a 2×3 matrix and matrix B is a 3×2 matrix, the resulting matrix will be a 2×2 matrix. The schematic representation of matrix addition is shown below

$$

\begin{bmatrix} a & b & c \\ d & e & f \end{bmatrix} + \begin{bmatrix} g & h \\ i & j \\ k & l \end{bmatrix} = \begin{bmatrix} (a+g)+(b+i)+(c+k) & (a+h)+(b+j)+(c+l) \\ (d+g)+(e+i)+(f+k) & (d+h)+(e+j)+(f+l) \end{bmatrix}

For instance,

$$

\begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix} + \begin{bmatrix} 7 & 8 \\ 9 & 10 \\ 11 & 12 \end{bmatrix} = \begin{bmatrix} 33 & 36 \\ 42 & 45 \end{bmatrix}

Note that the first matrix is a 2x3 matrix and the second matrix is a 3x2 matrix, the resulting matrix is a 2x2 matrix.

Suppose that you would like to compute the most likely outcome of the throws of two dice. Thus,

$$

\begin{bmatrix} 1 \\ 2 \\ 3 \\ 4 \\ 5 \\ 6 \end{bmatrix} + \begin{bmatrix} 1 & 2 & 3 & 4 & 5 & 6 \end{bmatrix} = \begin{bmatrix} 2 & 3 & 4 & 5 & 6 & 7\\ 3 & 4 & 5 & 6 & 7 & 8\\ 4 & 5 & 6 & 7 & 8 & 9\\ 5 & 6 & 7 & 8 & 9 & 10\\ 6 & 7 & 8 & 9 & 10 & 11\\ 7 & 8 & 9 & 10 & 11 & 12\\ \end{bmatrix}

The addition of matrices is especially useful for the visualization of higher transcendental functions in three dimensions, as shown below:

References

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• Krus, D.J., & Ceuvorst, R. W. (1979) Dominance, information, and hierarchical scaling of variance space. Applied Psychological Measurement, 3, 515-527.

• Krus, D.J., & Wilkinson, S.M. (1986) Matrix differencing as a concise expression of variance. Educational and Psychological Measurement, 46, 179-183. (Request reprint).
• Krus, D.J. (2002) Imaging higher transcendental functions in 3-Dimensions. Journal of Visual Statistics 1, 6-9.(Request reprint).

See also

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• matrix subtraction
• supermatrix

External links

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• Online matrix addition calculator
• Introduction to Matrix Algebra

• Matrix Algebra Operations on Matrices

Linear algebra

Addition matricielle | Somma fra matrici | Matrisaddition