Several types of mathematical matrix containing zeroes
In
mathematics, a hollow matrix may refer to one of several related classes of
matrix: a sparse matrix; a matrix with a large block of zeroes; or a matrix with diagonal entries all zero.
Definitions
Sparse
A hollow matrix may be one with "few" non-zero entries: that is, a
sparse matrix.[1]
Block of zeroes
A hollow matrix may be a square n × n matrix with an r × s block of zeroes where r + s > n.[2]
If A represents a
linear mapwith respect to a fixed
basis, then it maps each basis vector e into the
complement of the
span of e. That is, where
The
Gershgorin circle theorem shows that the moduli of the
eigenvalues of a hollow matrix are less or equal to the sum of the moduli of the non-diagonal row entries.
References
^Pierre Massé (1962). Optimal Investment Decisions: Rules for Action and Criteria for Choice.
Prentice-Hall. p. 142.