Hollow matrix

Résumé

In mathematics, a hollow matrix may refer to one of several related classes of matrix: a sparse matrix; a matrix with a large block of zeroes; or a matrix with diagonal entries all zero. A hollow matrix may be one with "few" non-zero entries: that is, a sparse matrix. A hollow matrix may be a square n × n matrix with an r × s block of zeroes where r + s > n. A hollow matrix may be a square matrix whose diagonal elements are all equal to zero. That is, an n × n matrix A = (aij) is hollow if aij = 0 whenever i = j (i.e. aii = 0 for all i). The most obvious example is the real skew-symmetric matrix. Other examples are the adjacency matrix of a finite simple graph, and a distance matrix or Euclidean distance matrix. In other words, any square matrix that takes the form is a hollow matrix, where the symbol denotes an arbitrary entry. For example, is a hollow matrix. The trace of a hollow matrix is zero. If A represents a linear map with respect to a fixed basis, then it maps each basis vector e into the complement of the span of e. That is, where . The Gershgorin circle theorem shows that the moduli of the eigenvalues of a hollow matrix are less or equal to the sum of the moduli of the non-diagonal row entries.

Source officielle

https://en.wikipedia.org/wiki/Hollow_matrix

À propos de ce résultat

Cette page est générée automatiquement et peut contenir des informations qui ne sont pas correctes, complètes, à jour ou pertinentes par rapport à votre recherche. Il en va de même pour toutes les autres pages de ce site. Veillez à vérifier les informations auprès des sources officielles de l'EPFL.