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| In [[mathematics]], a '''hollow matrix''' may refer to one of several related classes of [[matrix (mathematics)|matrix]].
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| ==Sparse==
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| A ''hollow matrix'' may be one with "few" non-zero entries: that is, a [[sparse matrix]].<ref>{{cite book | author=Pierre Massé | title=Optimal Investment Decisions: Rules for Action and Criteria for Choice | publisher=[[Prentice-Hall]] | year=1962 | page=142 }}</ref>
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| ==Diagonal entries all zero==
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| A ''hollow matrix'' may be a [[square matrix]] whose [[Diagonal#Matrices|diagonal]] elements are all equal to zero
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| .<ref>{{cite book | author=James E. Gentle | title=Matrix Algebra: Theory, Computations, and Applications in Statistics | publisher=[[Springer-Verlag]] | year=2007 | isbn=0-387-70872-3 | page=42 }}</ref> The most obvious example is the [[real numbers|real]] [[skew-symmetric matrix|skew-symmetric]] matrix. Other examples are the [[adjacency matrix]] of a finite [[simple graph]]; a [[distance matrix]] or [[Euclidean distance matrix]].
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| If ''A'' is an ''n''×''n'' hollow matrix, then the elements of ''A'' are given by
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| :<math>\begin{array}{rlll} | |
| A_{n\times n} & = & (a_{ij});
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| \\
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| a_{ij} & = & 0 & \mbox{if} \quad i=j,\quad 1\le i,j \le n.\,
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| \end{array}
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| </math>
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| In other words, any square matrix which takes the form <math>\left(\begin{array}{ccccc} 0\\ & 0\\ & & \ddots\\ & & & 0\\ & & & & 0\end{array}\right)</math> is a hollow matrix.
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| For example:
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| <math>\left(\begin{array}{ccccc} 0 & 2 & 6 & \frac{1}{3} & 4\\2 & 0 & 4 & 8 & 0\\ 9 & 4 & 0 & 2 & 933\\
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| 1 & 4 & 4 & 0 & 6\\ 7 & 9 & 23 & 8 & 0\end{array}\right)</math> is an example of a hollow matrix.
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| ===Properties=== | |
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| *The trace of ''A'' is trivially zero.
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| *The linear map represented by ''A'' (with respect to a fixed basis) maps each basis vector ''e'' onto the image of the complement of ''<e>''.
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| ==Block of zeroes==
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| A ''hollow matrix'' may be a square ''n''×''n'' matrix with an ''r''×''s'' block of zeroes where ''r''+''s''>''n''.<ref>{{cite book | author=Paul Cohn | authorlink=Paul Cohn | title=Free Ideal Rings and Localization in General Rings | publisher=[[Cambridge University Press]] | year=2006 | isbn=0-521-85337-0 | page=430 }}</ref>
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| ==References==
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| {{reflist}}
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| {{DEFAULTSORT:Hollow Matrix}}
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| [[Category:Matrices]]
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| {{Linear-algebra-stub}}
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