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If the row, or the column, or both, are ≡ 1 (mod 4) the entry is blue or green; if both row and column are ≡ 3 (mod 4), it is yellow or orange. The blue and green entries are symmetric around the diagonal: The entry for row p, column q is R (resp N) if and only if the entry at row q, column p, is R (resp N).
The dot product of two rows of the same type is congruent to n (mod 4); the dot product of two rows of opposite type is congruent to n+2 (mod 4). When n ≡ 2 (mod 4), this implies that, by permuting rows of R, we may assume the standard form, = [], where A and D are symmetric integer matrices whose elements are congruent to 2 (mod 4) and B is ...
Permutation matrix. In mathematics, particularly in matrix theory, a permutation matrix is a square binary matrix that has exactly one entry of 1 in each row and each column with all other entries 0. [1]: 26 An n × n permutation matrix can represent a permutation of n elements. Pre- multiplying an n -row matrix M by a permutation matrix P ...
In linear algebra, the rank of a matrix A is the dimension of the vector space generated (or spanned) by its columns. [1][2][3] This corresponds to the maximal number of linearly independent columns of A. This, in turn, is identical to the dimension of the vector space spanned by its rows. [4] Rank is thus a measure of the "nondegenerateness ...
The rows of the inverse matrix V of a matrix U are orthonormal to the columns of U (and vice versa interchanging rows for columns). To see this, suppose that UV = VU = I where the rows of V are denoted as v i T {\displaystyle v_{i}^{\mathrm {T} }} and the columns of U as u j {\displaystyle u_{j}} for 1 ≤ i , j ≤ n . {\displaystyle 1\leq i,j ...
Weighing matrices are so called because of their use in optimally measuring the individual weights of multiple objects. [1][2] In mathematics, a weighing matrix of order and weight is a matrix with entries from the set such that: Where is the transpose of and is the identity matrix of order . The weight is also called the degree of the matrix.
A set of 20 points in a 10 × 10 grid, with no three points in a line. The no-three-in-line problem in discrete geometry asks how many points can be placed in the grid so that no three points lie on the same line. The problem concerns lines of all slopes, not only those aligned with the grid. It was introduced by Henry Dudeney in 1900.
E.g. the last row is computed as follows: If you insert = in the equation x 3 + x + 1 mod 5 you get as result (3rd column). This result can be achieved if =, (Quadratic residues can be looked up in the 2nd column). So the points for the last row are (,), (,).