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

$\begin{array}{rll}{A}_{n\times n}& =& (a_{ij});\\ a_{ij}& =& 0& \text{if}i=j,1\le i,j\le n.\end{array}$

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