Matrix equivalence


Matrix equivalence

In linear algebra, two rectangular m-by-n matrices A and B are called equivalent if

\! B = Q^{-1} A P

for some invertible n-by-n matrix P and some invertible m-by-m matrix Q. Equivalent matrices represent the same linear transformation V → W under two different choices of a pair of bases of V and W, with P and Q being the change of basis 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 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.
  • The matrices have the same rank.

See also


Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Matrix - получить на Академике рабочий купон на скидку Летуаль или выгодно matrix купить с бесплатной доставкой на распродаже в Летуаль

  • Matrix congruence — In mathematics, two matrices A and B over a field are called congruent if there exists an invertible matrix P over the same field such that PTAP = B where T denotes the matrix transpose. Matrix congruence is an equivalence relation. Matrix… …   Wikipedia

  • Matrix norm — In mathematics, a matrix norm is a natural extension of the notion of a vector norm to matrices. Contents 1 Definition 2 Induced norm 3 Entrywise norms 3.1 Frobenius norm …   Wikipedia

  • Matrix ring — In abstract algebra, a matrix ring is any collection of matrices forming a ring under matrix addition and matrix multiplication. The set of n×n matrices with entries from another ring is a matrix ring, as well as some subsets of infinite matrices …   Wikipedia

  • Equivalence class — This article is about equivalency in mathematics; for equivalency in music see equivalence class (music). In mathematics, given a set X and an equivalence relation on X, the equivalence class of an element a in X is the subset of all elements in… …   Wikipedia

  • Matrix mechanics — Quantum mechanics Uncertainty principle …   Wikipedia

  • Morita equivalence — In abstract algebra, Morita equivalence is a relationship defined between rings that preserves many ring theoretic properties. It is named after Japanese mathematician Kiiti Morita who defined equivalence and a similar notion of duality in 1958.… …   Wikipedia

  • 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 times; n matrices are row equivalent if and only if they have the same row space. The concept… …   Wikipedia

  • Non-negative matrix factorization — NMF redirects here. For the bridge convention, see new minor forcing. Non negative matrix factorization (NMF) is a group of algorithms in multivariate analysis and linear algebra where a matrix, , is factorized into (usually) two matrices, and… …   Wikipedia

  • Hadamard matrix — In mathematics, a Hadamard matrix is a square matrix whose entries are either +1 or −1 and whose rows are mutually orthogonal. This means that every two different rows in a Hadamard matrix represent two perpendicular vectors. Such matrices can… …   Wikipedia

  • Band matrix — In mathematics, particularly matrix theory, a band matrix is a sparse matrix whose non zero entries are confined to a diagonal band, comprising the main diagonal and zero or more diagonals on either side. Contents 1 Matrix bandwidth 2… …   Wikipedia


We are using cookies for the best presentation of our site. Continuing to use this site, you agree with this.