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In mathematics, a block matrix or a partitioned matrix is a matrix that is interpreted as having been broken into sections called blocks or submatrices. [1] [2]Intuitively, a matrix interpreted as a block matrix can be visualized as the original matrix with a collection of horizontal and vertical lines, which break it up, or partition it, into a collection of smaller matrices.
Synonym for kernel-Hermitian matrices. Examples include (but not limited) to Hermitian, skew-Hermitian matrices, and normal matrices. Partitioned matrix: A matrix partitioned into sub-matrices, or equivalently, a matrix whose entries are themselves matrices rather than scalars. Synonym for block matrix. Parisi matrix: A block-hierarchical matrix.
partition of a matrix; see block matrix, and partition of the sum of squares in statistics problems, especially in the analysis of variance , quotition and partition , two ways of viewing the operation of division of integers.
(Note: this is the partition, not a member of the partition.) For any non-empty set X, P = { X} is a partition of X, called the trivial partition. Particularly, every singleton set {x} has exactly one partition, namely { {x} }. For any non-empty proper subset A of a set U, the set A together with its complement form a partition of U, namely ...
Two important examples are the partitions restricted to only odd integer parts or only even integer parts, with the corresponding partition functions often denoted () and (). A theorem from Euler shows that the number of strict partitions is equal to the number of partitions with only odd parts: for all n , q ( n ) = p o ( n ) {\displaystyle q ...
Denote the n objects to partition by the integers 1, 2, ..., n. Define the reduced Stirling numbers of the second kind, denoted (,), to be the number of ways to partition the integers 1, 2, ..., n into k nonempty subsets such that all elements in each subset have pairwise distance at least d.
In mathematics, the Khatri–Rao product or block Kronecker product of two partitioned matrices and is defined as [1] [2] [3] = in which the ij-th block is the m i p i × n j q j sized Kronecker product of the corresponding blocks of A and B, assuming the number of row and column partitions of both matrices is equal.
Graph partition into subgraphs of specific types (triangles, isomorphic subgraphs, Hamiltonian subgraphs, forests, perfect matchings) are known NP-complete. Partition into cliques is the same problem as coloring the complement of the given graph. A related problem is to find a partition that is optimal terms of the number of edges between parts ...