linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrice ...

, two rectangular ''m''-by-''n''

matrices Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...

''A'' and ''B'' are called equivalent if :

B = Q^ A P

for some

invertible In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers. Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that is ...

''n''-by-''n'' matrix ''P'' and some invertible ''m''-by-''m'' matrix ''Q''. Equivalent matrices represent the same

linear transformation In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...

''V'' → ''W'' under two different choices of a pair of bases of ''V'' and ''W'', with ''P'' and ''Q'' being the

change of basis In mathematics, an ordered basis of a vector space of finite dimension allows representing uniquely any element of the vector space by a coordinate vector, which is a sequence of scalars called coordinates. If two different bases are consider ...

matrices in ''V'' and ''W'' respectively. The notion of equivalence should not be confused with that of similarity, which is only defined for square matrices, and is much more restrictive (similar matrices are certainly equivalent, but equivalent square matrices need not be similar). That notion corresponds to matrices representing the same

endomorphism In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space is a linear map , and an endomorphism of a gr ...

''V'' → ''V'' under two different choices of a ''single'' basis of ''V'', used both for initial vectors and their images.

Properties

Matrix equivalence is an

equivalence relation In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relatio ...

on the space of rectangular matrices. For two rectangular matrices of the same size, their equivalence can also be characterized by the following conditions * The matrices can be transformed into one another by a combination of elementary row and column operations. * Two matrices are equivalent if and only if they have the same

rank Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as: Level or position in a hierarchical organization * Academic rank * Diplomatic rank * Hierarchy * ...

Canonical form

The

property yields an intuitive

canonical form In mathematics and computer science, a canonical, normal, or standard form of a mathematical object is a standard way of presenting that object as a mathematical expression. Often, it is one which provides the simplest representation of an ...

for matrices of the equivalence class of rank

k

\begin
1 & 0 & 0 & & \cdots & & 0 \\
0 & 1 & 0 & & \cdots & & 0 \\
0 & 0 & \ddots & & & & 0\\
\vdots & & & 1 & & & \vdots \\
 & & & & 0 & & \\
 & & & & & \ddots &  \\
0 & & & \cdots & & & 0
\end

, where the number of

1

s on the diagonal is equal to

k

. This is a special case of the Smith normal form, which generalizes this concept on vector spaces to free modules over principal ideal domains.

Properties

Canonical form

See also