have the same null space. Because elementary row operations are reversible, row equivalence is an equivalence relation. It is commonly denoted by a tilde...

matrix equivalent are not necessarily row equivalent. This makes matrix equivalence a generalization of row equivalence. The rank property yields an intuitive...

additivity formula Matrix equivalence Matrix congruence Matrix similarity Matrix consimilarity Row equivalence Elementary row operations Householder transformation...

In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric, and transitive. The equipollence relation between line segments...

exactly one subset. Every equivalence relation on a set defines a partition of this set, and every partition defines an equivalence relation. A set equipped...

Enharmonic equivalence In music theory, equivalence class is an equality (=) or equivalence between properties of sets (unordered) or twelve-tone rows (ordered...

b = g a g − 1 . {\displaystyle b=gag^{-1}.} This is an equivalence relation whose equivalence classes are called conjugacy classes. In other words, each...

the columns of C. It follows from the equivalence ( 1 ) ⇔ ( 5 ) {\displaystyle (1)\Leftrightarrow (5)} that the row rank is equal to the column rank. As...

isotopic to a conjugate of the other. This is again an equivalence relation, with the equivalence classes called main classes, species, or paratopy classes...

preorder is an equivalence relation. Moreover, a preorder on a set X {\displaystyle X} can equivalently be defined as an equivalence relation on X {\displaystyle...

monoid of all words in the alphabet of positive integers modulo Knuth equivalence. Its elements can be identified with semistandard Young tableaux. It...

by transposition. It is similar to enharmonic equivalence, octave equivalence, and inversional equivalence. In many musical contexts, transpositionally...

Exclusive or, exclusive disjunction, exclusive alternation, logical non-equivalence, or logical inequality is a logical operator whose negation is the logical...

automorphic equivalence, and regular equivalence. There is a hierarchy of the three equivalence concepts: any set of structural equivalences are also automorphic...

begin with zero. More generally, for a class of objects on which an equivalence relation is defined, a canonical form consists in the choice of a specific...

the logical biconditional, also known as material biconditional or equivalence or bidirectional implication or biimplication or bientailment, is the...

"Dram" equivalence is sometimes still used as a measure of the powder charge power in a cartridge. Today, it is an anachronistic equivalence that represents...

defined as their dot product. So the equivalence of the two definitions of the dot product is a part of the equivalence of the classical and the modern formulations...

into account). There are 9,985,920 classes of twelve-tone rows up to equivalence (where two rows are equivalent if one is a transformation of the other)...

the conditional. Truth tables can be used to prove many other logical equivalences. For example, consider the following truth table: This demonstrates the...

is called the language recognized by A; it is denoted as L(A). For an equivalence relation ≈ on the set S of A’s states, the quotient automaton A/≈ = ⟨Σ...

n-dimensional manifolds M and N is an h-cobordism (the h stands for homotopy equivalence) if the inclusion maps M ↪ W and N ↪ W {\displaystyle M\hookrightarrow...

connectives include negation, disjunction, conjunction, implication, and equivalence. In standard systems of classical logic, these connectives are interpreted...

if x−1y belongs to H). The equivalence classes of this relation are the left cosets of H. As with any set of equivalence classes, they form a partition...

and b to c, then R also relates a to c. Every partial order and every equivalence relation is transitive. For example, less than and equality among real...

{\displaystyle [x:y]} for homogeneous coordinates denotes the equivalence class of the row vector. The elements of GL2(R) that act trivially on P1(R) are...

Cartesian product of rows from tables in the join. In other words, it will produce rows which combine each row from the first table with each row from the second...

twelve-tone technique, derivation is the construction of a row through segments. A derived row is a tone row whose entirety of twelve tones is constructed from...

pitch class is "all pitches related to each other by octave, enharmonic equivalence, or both." Thus, using scientific pitch notation, the pitch class "C"...

symmetry and transitivity, reflexivity is one of three properties defining equivalence relations. The word reflexive is originally derived from the Medieval...