Hollow matrix

From Wikipedia, the free encyclopedia

In mathematics, a hollow matrix is an n×n (i.e. square) matrix whose diagonal elements are all equal to zero. The most obvious example is the real skew-symmetric matrix.

If A is an n×n hollow matrix, then the elements of A are given by

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

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.