The operations on matrices differ from similar operations of scalar algebra in several respects. The matrix algebra operations, in general, are not commutative and attention must be paid to whether the matrices are conformable with respect to the intended operation. Also, it must be noted whether the matrix operation pertains to matrix elements or to matrices.

Addition of matrix elements

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, i.e., C = A + B. For example For the square matrix section, see square matrix. ...

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

For instance,

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 abstract algebra, the direct sum is a construction which combines several vector spaces (or groups, or abelian groups, or modules) into a new, bigger one. ... A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. ...

Addition of matrices

Textbooks on matrix algebra, routinely describing major and minor vector products, do not suggest analogical operations for major and minor sums. These operations are easy to imagine and are not discussed because most of their potential applications can be as well accomplished by multiplications using the unit vectors. However, on close scrutiny, matrix algebra operations of addition (of vectors, not elements of vectors, of matrices, not elements of matrices) 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 another 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 3×2 matrix and matrix B is a 2×3 matrix, the resulting matrix will be a 2×2 matrix. The schematic representation of matrix addition is shown below

For instance,

Note that the first matrix is a 2x3 matrix and the second matrix is a 3x2 matrix, the resulting matrix is a 2x2 matrix. The addition of matrices is especially useful, e.g., for the visualization of higher transcendental functions in three dimensions, as shown below:

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References

• 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.
• Krus, D.J. (2002) Imaging higher transcendental functions in 3-Dimensions. Journal of Visual Statistics 1, 6-9. (Request reprint).

See also

• matrix subtraction
• supermatrix

// Introduction The operations on matrices differ from similar operations of scalar algebra in several respects. ... A supermatrix is a matrix consisting of submatrices. ...

External links

• Online matrix addition calculator
• Introduction to Matrix Algebra

• Matrix Operations on Matrix Elements

• Matrix Operations on Matrices