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For functions in certain classes, the problem of determining: whether two functions are equal, known as the zero-equivalence problem (see Richardson's theorem); [5] the zeroes of a function; whether the indefinite integral of a function is also in the class. [6] Of course, some subclasses of these problems are decidable.
Given an n × n square matrix A of real or complex numbers, an eigenvalue λ and its associated generalized eigenvector v are a pair obeying the relation [1] =,where v is a nonzero n × 1 column vector, I is the n × n identity matrix, k is a positive integer, and both λ and v are allowed to be complex even when A is real.l When k = 1, the vector is called simply an eigenvector, and the pair ...
A permutation matrix is a (0, 1)-matrix, all of whose columns and rows each have exactly one nonzero element.. A Costas array is a special case of a permutation matrix.; An incidence matrix in combinatorics and finite geometry has ones to indicate incidence between points (or vertices) and lines of a geometry, blocks of a block design, or edges of a graph.
In fact, it is already undecidable for sets of 6 matrices (or more) when n = 3, for 4 matrices when n = 5, for 3 matrices when n = 9, and for 2 matrices when n = 15 [2]. In the case n = 2, it is an open problem whether matrix mortality is decidable, but several special cases have been solved: the problem is decidable for sets of 2 matrices [ 3 ...
The orthogonal Procrustes problem [1] is a matrix approximation problem in linear algebra. In its classical form, one is given two matrices A {\displaystyle A} and B {\displaystyle B} and asked to find an orthogonal matrix Ω {\displaystyle \Omega } which most closely maps A {\displaystyle A} to B {\displaystyle B} .
Vector logic [1] [2] is an algebraic model of elementary logic based on matrix algebra. Vector logic assumes that the truth values map on vectors, and that the monadic and dyadic operations are executed by matrix operators. "Vector logic" has also been used to refer to the representation of classical propositional logic as a vector space, [3 ...
A bistochastic matrix (also called: doubly-stochastic) is a matrix in which all elements are greater than or equal to 0 and the sum of the elements in each row and column equals 1. An example is the following 3-by-3 matrix: A permutation matrix is a special case of a bistochastic matrix, in which each element is either 0 or 1 (so there is ...
In theoretical computer science, the computational complexity of matrix multiplication dictates how quickly the operation of matrix multiplication can be performed. Matrix multiplication algorithms are a central subroutine in theoretical and numerical algorithms for numerical linear algebra and optimization, so finding the fastest algorithm for matrix multiplication is of major practical ...