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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.
Then the partial trace of , with respect to either system A or B, is a diagonal matrix whose non-zero diagonal elements are | |. In other words, the Schmidt decomposition shows that the reduced states of ρ {\displaystyle \rho } on either subsystem have the same spectrum.
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.
More generally, we can factor a complex m×n matrix A, with m ≥ n, as the product of an m×m unitary matrix Q and an m×n upper triangular matrix R.As the bottom (m−n) rows of an m×n upper triangular matrix consist entirely of zeroes, it is often useful to partition R, or both R and Q:
(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 ...
In general, if the integer n is partitioned into a sum in which "1" appears j 1 times, "2" appears j 2 times, and so on, then the number of partitions of a set of size n that collapse to that partition of the integer n when the members of the set become indistinguishable is the corresponding coefficient in the polynomial.
The partition on the right-hand side should be the transpose of the partition on the left-hand side, in the sense that if A is an m-by-n block then E should be an n-by-m block. The statement of the nullity theorem is now that the nullities of the blocks on the right equal the nullities of the blocks on the left ( Strang & Nguyen 2004 ):