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The value of the rank function for an element of the poset is called its rank. Sometimes a graded poset is called a ranked poset but that phrase has other meanings; see Ranked poset. A rank or rank level of a graded poset is the subset of all the elements of the poset that have a given rank value. [1] [2]
A matrix that has rank min(m, n) is said to have full rank; otherwise, the matrix is rank deficient. Only a zero matrix has rank zero. f is injective (or "one-to-one") if and only if A has rank n (in this case, we say that A has full column rank). f is surjective (or "onto") if and only if A has rank m (in this case, we say that A has full row ...
There are several known constructions of rank codes, which are maximum rank distance (or MRD) codes with d = n − k + 1.The easiest one to construct is known as the (generalized) Gabidulin code, it was discovered first by Delsarte (who called it a Singleton system) and later by Gabidulin [2] (and Kshevetskiy [3]).
In practice, we can construct one specific rank factorization as follows: we can compute , the reduced row echelon form of .Then is obtained by removing from all non-pivot columns (which can be determined by looking for columns in which do not contain a pivot), and is obtained by eliminating any all-zero rows of .
If G is a finite non-abelian simple group (e.g. G = A n, the alternating group, for n > 4) then rank(G) = 2. This fact is a consequence of the Classification of finite simple groups . If G is a finitely generated group and Φ( G ) ≤ G is the Frattini subgroup of G (which is always normal in G so that the quotient group G /Φ( G ) is defined ...
Other surprising examples include torsion-free rank 2 groups A n,m and B n,m such that A n is isomorphic to B n if and only if n is divisible by m. For abelian groups of infinite rank, there is an example of a group K and a subgroup G such that K is indecomposable; K is generated by G and a single other element; and
There is exactly one zero matrix of any given dimension m×n (with entries from a given ring), so when the context is clear, one often refers to the zero matrix. In general, the zero element of a ring is unique, and is typically denoted by 0 without any subscript indicating the parent ring. Hence the examples above represent zero matrices over ...
For a square N×N matrix A n,m = A(n,m), in-place transposition is easy because all of the cycles have length 1 (the diagonals A n,n) or length 2 (the upper triangle is swapped with the lower triangle). Pseudocode to accomplish this (assuming zero-based array indices) is: for n = 0 to N - 1 for m = n + 1 to N swap A(n,m) with A(m,n)