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Krus, D. J. (2001) Matrix addition. Journal of Visual Statistics, 1, (February, 2001).
Matrix addition
David J. Krus
Arizona State University
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 addition of matrix elements is defined for matrices of same dimensions and
is computed by adding their corresponding elements, i.e.,
C = A (+) B, as
For instance,
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. The schematic representation of matrix addition
is shown below
For instance,
Note that the first matrix is a 3x2 matrix and the second matrix is a 2x3 matrix, the resulting matrix is a 3x3 matrix. The matrix addition is a useful operation with many applications. Suppose that you would like to compute the most likely outcome of the throws of two dice. You may add the possible outcomes of a throw of a dice as
and get your answer by just
looking on the results. The addition of matrices is especially useful for the
visualization of objects in the three dimensions, as, e.g., the visualization
of higher transcendental functions in three dimensions, as shown below.
References
Krus, D. J., & Ceuvorst, R. W. (1979) Dominance, information, and hierarchical scaling of variance space. Applied Psychological Measurement, 3, 515-527 (Request reprint).
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 at www.visualstatistics.net (March 14) (Request reprint).
See
also