The nilpotent-centralizer method for spectrally arbitrary patterns

Techniques for identifying inertially arbitrary patterns

“…Conversely, if Jac(f )| A has rank n, then by Theorem 4.3, the set The centralizer of a matrix A is the set of all matrices that commute with A. Analogous to Lemma 3.6 in [9], we have the following lemma. Proof.…”

“…In [8], the nilpotent-Jacobian method is developed to show a pattern A is spectrally arbitrary by first finding a nilpotent matrix A ∈ Q(A) and then verifying that a certain Jacobian matrix is nonsingular so that the Implicit Function Theorem can be applied. In [9], the generalized n×m Jacobian matrix for a matrix A of order n with m nonzero entries is discussed and the nilpotent-centralizer method is developed using an algebraic property of a certain nilpotent matrix to show a pattern is spectrally arbitrary. In the proofs of both the nilpotent-Jacobian method and the nilpotent-centralizer method, a nilpotent matrix is critical.…”

“…Generalizations of the nilpotent-centralizer method. In this section, we extend the nilpotent-centralizer method, recently developed in [9] for showing that a pattern is spectrally arbitrary, to give other techniques to show that a pattern is inertially arbitrary.…”

“…The following lemma from [9] describes the Jacobian matrix of the polynomial map g of a square matrix A in terms of the adjugate of zI − A. For a matrix M , the notation M ij denotes its (i, j) entry.…”

“…Note that if a z = n in Theorem 4.9, then every superpattern of A is in fact spectrally arbitrary [9]. However if a z < n, this is not in general true; see, for example, W n in Section 4.1 and G n with n = 2k + 1 in Section 4.3, which do not allow refined inertia (0, 0, n, 0), and thus are not spectrally arbitrary.…”