On the nonsingularity of complex matrices

On a problem of O. Taussky

“…These results are described in § 3. Also in this section is a generalization (Theorems 3 and 4) of a recent interesting result by Camion and Hoffman [1]. Our proof of this generalization differs from that of [1].…”

“…We remark that the special case σ = 0 of Theorems 3 and 4 corresponds to the main results of [1], Letting R f denote the complement of any set R in the complex plane, then Theorem 4 implies: We now consider the closed connected components of S(Ωj). We remark that, since (S(Ω Λ )y always contains the unbounded connected component {z | | z | > p(A)}, the identity permutation must always occur as one of the r permutations of Theorem 6.…”

“…In this case, there is evidently a repeated first index, and for convenience, we assume that 1 = j(T) -j(2) = = j(r), r ^ 2. Then let y be any nonnegative vector with 7/1 + 2/2 = 1, 2/y = 0 for 2 < j ^ n. For such vectors, it follows from (3.9) that We can further show that, if σ g S(ί2J, then as in [1] there is a unique permutation φ such that | σ \ g G φ {Ω A <p). This will permit us to show that at most (n + 1) permutations are necessary to characterize…”

“…In [3], the authors completely characterized the spectrum S(Ω σ ) of a related set Ω o of matrices, where C = (c itj ) was an arbitrary n x 7i complex matrix and (1. (1)(2)(3)(4) S(Ω A ) = U S(Ω Λmθ) ) .…”

Localization of the Zeros of the Permanent of a Characteristic Matrix

“…As Camion and Hoffman [4] have shown, if det B 5^ 0 for every BexY{A), then there exist a permutation matrix P and a nonsingular diagonal matrix D such that PAD is a dominant diagonal matrix. Brenner [1] has shown that the permanent of a dominant diagonal matrix is nonzero.…”