Three observations on the determinantal range
Graça Soares2005, Linear Algebra and its Applications
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Abstract
Let A, C ∈ M n , the algebra of n × n complex matrices. The set of complex numbers C (A) = {det (A − U CU *) : U * U = I n } is the C-determinantal range of A. In this note, it is proved that C (A) is an elliptical disc for A, C ∈ M 2. A necessary and sufficient condition for C (A) to be a line segment is given when A and C are normal matrices with pairwise distinct eigenvalues. The linear operators L that satisfy the linear preserver property C (A) = C (L(A)), for all A, C ∈ M n , are characterized.
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