Hollow matrix

Hollow matrix

In mathematics, a hollow matrix may refer to one of several related classes of matrix.

parse

A "hollow matrix" may be one with "few" non-zero entries: that is, a sparse matrix [cite book | author=Pierre Massé | title=Optimal Investment Decisions: Rules for Action and Criteria for Choice | publisher=Prentice-Hall | date=1962 | page=142 ] .

Diagonal entries all zero

A "hollow matrix" may be a square matrix whose diagonal elements are all equal to zero [cite book | author=Jon Dattorro | title=Convex Optimization and Euclidean Distance Geometry | date=2005 | publisher=Meboo | isbn=0976401304 | url=http://www.stanford.edu/~dattorro/mybook.html | page=65 ] [cite book | author=James E. Gentle | title=Matrix Algebra: Theory, Computations, and Applications in Statistics | publisher=Springer-Verlag | date=2007 | isbn=0387708723 | page=42 ] . The most obvious example is the real skew-symmetric matrix. Other examples are the adjacency matrix of a finite simple graph; a distance matrix or Euclidean distance matrix.

If "A" is an "n"×"n" hollow matrix, then the elements of "A" are given by

:egin{array}{rlll}A_{n imes n} & = & (a_{ij});\a_{ij} & = & 0 & mbox{if} quad i=j,quad 1le i,j le n.,end{array}

In other words, any square matrix which takes the form left(egin{array}{ccccc} 0\ & 0\ & & ddots\ & & & 0\ & & & & 0end{array} ight) is a hollow matrix.

For example:left(egin{array}{ccccc} 0 & 2 & 6 & frac{1}{3} & 4\2 & 0 & 4 & 8 & 0\ 9 & 4 & 0 & 2 & 933\1 & 4 & 4 & 0 & 6\ 7 & 9 & 23 & 8 & 0end{array} ight) is an example of a hollow matrix.

Properties

*The trace of "A" is trivially zero.

Block of zeroes

A "hollow matrix" may be a square "n"×"n" matrix with an "r"×"s" block of zeroes where "r"+"s">"n". [cite book | author=Paul Cohn | authorlink=Paul Cohn | title=Free Ideal Rings and Localization in General Rings | publisher=Cambridge University Press | date=2006 | isbn=0-521-85337-0 | page=430 ]

Properties

* Such a matrix is necessarily singular.

References


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