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The identity matrices (which are the square matrices whose entries are zero outside of the main diagonal and 1 on the main diagonal) are identity elements of the matrix product. It follows that the n × n matrices over a ring form a ring, which is noncommutative except if n = 1 and the ground ring is commutative.
A special orthogonal matrix is an orthogonal matrix with determinant +1. As a linear transformation, every orthogonal matrix with determinant +1 is a pure rotation without reflection, i.e., the transformation preserves the orientation of the transformed structure, while every orthogonal matrix with determinant -1 reverses the orientation, i.e ...
The best known [1] general exact algorithm is due to H. J. Ryser ().Ryser's method is based on an inclusion–exclusion formula that can be given [2] as follows: Let be obtained from A by deleting k columns, let () be the product of the row-sums of , and let be the sum of the values of () over all possible .
For matrix-matrix exponentials, there is a distinction between the left exponential Y X and the right exponential X Y, because the multiplication operator for matrix-to-matrix is not commutative. Moreover, If X is normal and non-singular, then X Y and Y X have the same set of eigenvalues. If X is normal and non-singular, Y is normal, and XY ...
For m = 0, A and B are empty matrices (but of different shapes if n > 0), as is their product AB; the summation involves a single term S = Ø, and the formula states 1 = 1, with both sides given by the determinant of the 0×0 matrix. For m = 1, the summation ranges over the collection ([]) of the n different singletons taken from [n], and both ...
An n-by-n square matrix A is called invertible (also nonsingular, nondegenerate or rarely regular) if there exists an n-by-n square matrix B such that = =, where I n denotes the n-by-n identity matrix and the multiplication used is ordinary matrix multiplication. [1]
Let Ω(n,k) be the class of all (0, 1)-matrices of order n with each row and column sum equal to k. Every matrix A in this class has perm(A) > 0. [13] The incidence matrices of projective planes are in the class Ω(n 2 + n + 1, n + 1) for n an integer > 1. The permanents corresponding to the smallest projective planes have been calculated.
The set of all doubly stochastic matrices is called the Birkhoff polytope, and the permutation matrices play a special role in that polytope. The Birkhoff–von Neumann theorem says that every doubly stochastic real matrix is a convex combination of permutation matrices of the same order, with the permutation matrices being precisely the ...