Matrix Equivalence
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In
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 matrices. ...
, 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 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
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 * ...
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 as \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 1s on the diagonal is equal to k. This is a special case of the
Smith normal form In mathematics, the Smith normal form (sometimes abbreviated SNF) is a normal form that can be defined for any matrix (not necessarily square) with entries in a principal ideal domain (PID). The Smith normal form of a matrix is diagonal, and can b ...
, which generalizes this concept on vector spaces to free modules over principal ideal domains.


See also

*
Matrix similarity In linear algebra, two ''n''-by-''n'' matrices and are called similar if there exists an invertible ''n''-by-''n'' matrix such that B = P^ A P . Similar matrices represent the same linear map under two (possibly) different bases, with being ...
*
Row equivalence In linear algebra, two matrices are row equivalent if one can be changed to the other by a sequence of elementary row operations. Alternatively, two ''m'' × ''n'' matrices are row equivalent if and only if they have the same row ...
*
Matrix congruence In mathematics, two Square matrix, square matrices ''A'' and ''B'' over a Field (mathematics), field are called congruent if there exists an invertible matrix ''P'' over the same field such that :''P''T''AP'' = ''B'' where "T" denotes the matri ...
{{Matrix classes Matrices Equivalence (mathematics)