In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

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

Matrix (mathematics)

In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...

.

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

Skew-symmetric matrix

In mathematics, and in particular linear algebra, a skew-symmetric matrix is a square matrix A whose transpose is also its negative; that is, it satisfies the equation If the entry in the and is aij, i.e...

 matrix. Other examples are the adjacency matrix

Adjacency matrix

In mathematics and computer science, an adjacency matrix is a means of representing which vertices of a graph are adjacent to which other vertices...

 of a finite simple graph; a distance matrix

Distance matrix

In mathematics, computer science and graph theory, a distance matrix is a matrix containing the distances, taken pairwise, of a set of points...

 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 .

Block of zeroes

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