Imaginary line (mathematics)

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.

Diagonal entries all zero

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

${\begin{array}{rlll}A_{n\times n}&=&(a_{ij});\\a_{ij}&=&0&{\mbox{if}}\quad i=j,\quad 1\leq i,j\leq n.\,\end{array}}$

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

For example: $\left({\begin{array}{ccccc}0&2&6&{\frac {1}{3}}&4\\2&0&4&8&0\\9&4&0&2&933\\1&4&4&0&6\\7&9&23&8&0\end{array}}\right)$  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.

References

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