- 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 theadjacency matrix of a finitesimple graph ; adistance matrix orEuclidean distance matrix .If "A" is an "n"×"n" hollow matrix, then the elements of "A" are given by
:
In other words, any square matrix which takes the form is a hollow matrix.
For example: 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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