Properties and preservers of the pseudospectrum

The interplay between the algebraic and analytic properties of a matrix and the geometric properties of its pseudospectrum is investigated. It is shown that one can characterize Hermitian matrices, positive semi-definite matrices, orthogonal projections, unitary matrices, etc. in terms of the pseudospectrum. Also, characterizations are given to maps on matrices leaving invariant the pseudospectrum of the sum, difference, or product of matrix pairs. It is shown that such a map is always a unitary similarity transform followed by some simple operations such as adding a constant matrix, taking the matrix transpose, or multiplying by a scalar in {1,-1}.