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In [[mathematics]], a '''hollow matrix''' may refer to one of several related classes of [[matrix (mathematics)|matrix]].
 
==Sparse==
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>
 
==Diagonal entries all zero==
A ''hollow matrix'' may be a [[square matrix]] whose [[Diagonal#Matrices|diagonal]] elements are all equal to zero
.<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]].
 
If ''A'' is an ''n''×''n'' hollow matrix, then the elements of ''A'' are given by
 
:<math>\begin{array}{rlll}
A_{n\times n} & = & (a_{ij});
\\
a_{ij} & = & 0 & \mbox{if} \quad i=j,\quad 1\le i,j \le n.\,
\end{array}
</math>
 
In other words, any square matrix which takes the form <math>\left(\begin{array}{ccccc} 0\\ & 0\\ &  & \ddots\\ &  &  & 0\\ &  &  &  & 0\end{array}\right)</math>&nbsp; is a hollow matrix.
 
For example:
<math>\left(\begin{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 & 0\end{array}\right)</math>&nbsp; is an example of a hollow matrix.
 
===Properties===
 
*The trace of ''A'' is trivially zero.
 
*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>''.
 
==Block of zeroes==
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>
 
==References==
{{reflist}}
 
{{DEFAULTSORT:Hollow Matrix}}
[[Category:Matrices]]
 
 
{{Linear-algebra-stub}}

Revision as of 19:08, 1 June 2013

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

Sparse

A hollow matrix may be one with "few" non-zero entries: that is, a sparse matrix.[1]

Diagonal entries all zero

A hollow matrix may be a square matrix whose diagonal elements are all equal to zero .[2] 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

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.
  • 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>.

Block of zeroes

A hollow matrix may be a square n×n matrix with an r×s block of zeroes where r+s>n.[3]

References

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Template:Linear-algebra-stub

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  2. 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

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  3. 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

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