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There is a simple generalisation to matrices with more columns and rows such that the i th row sum is equal to r i (a positive integer), the column sums are equal to 1, and all cells are non-negative (the sum of the row sums being equal to the number of columns). Any matrix in this form can be expressed as a convex combination of matrices in ...
Two matrices must have an equal number of rows and columns to be added. [1] In which case, the sum of two matrices A and B will be a matrix which has the same number of rows and columns as A and B. The sum of A and B, denoted A + B, is computed by adding corresponding elements of A and B: [2] [3]
A substochastic matrix is a real square matrix whose row sums are all ; In the same vein, one may define a probability vector as a vector whose elements are nonnegative real numbers which sum to 1. Thus, each row of a right stochastic matrix (or column of a left stochastic matrix) is a probability vector.
A matrix with the same number of rows and columns is called a square matrix. [5] A matrix with an infinite number of rows or columns (or both) is called an infinite matrix. In some contexts, such as computer algebra programs, it is useful to consider a matrix with no rows or no columns, called an empty matrix.
Given three matrices A, B and C, the products (AB)C and A(BC) are defined if and only if the number of columns of A equals the number of rows of B, and the number of columns of B equals the number of rows of C (in particular, if one of the products is defined, then the
For example, the boolean sum (that is, the bitwise OR) of the first two columns is =; that sum is not attainable as the sum of any other pair of columns in the matrix. However, this matrix is not 3-separable, because the sum of columns 1, 2, and 3 (namely 111111 {\displaystyle 111111} ) equals the sum of columns 1, 4, and 5.
A matrix where each row is a circular shift of its predecessor. Conference matrix: A square matrix with zero diagonal and +1 and −1 off the diagonal, such that C T C is a multiple of the identity matrix. Complex Hadamard matrix: A matrix with all rows and columns mutually orthogonal, whose entries are unimodular. Compound matrix
A row can be replaced by the sum of that row and a multiple of another row. R i + k R j → R i , where i ≠ j {\displaystyle R_{i}+kR_{j}\rightarrow R_{i},{\mbox{where }}i\neq j} If E is an elementary matrix, as described below, to apply the elementary row operation to a matrix A , one multiplies A by the elementary matrix on the left, EA .