On the crossing rule

On spaces of matrices containing a nonzero matrix of bounded rank

“…Friedland, Robbin and Sylvester [8] and Berger and Friedland [5] gave further nonlinear versions of the above results by considering odd maps φ from S p to matrices of rank n in M n (R), S n (R) and M n,n+1 (R) respectively. In this paper ( §4) we generalize these results to odd maps from S p to rank n − 1 matrices in M n (R) and S n (R).…”

“…See the works [13], [11], [2], [8], [5], [3], [4], [7] and many others. Roughly speaking these problems are divided into two classes depending on whether F is algebraically closed or not.…”

The Formulation and Analysis of Numerical Methods for Inverse Eigenvalue Problems

“…However, the principles on which these ideas rest are well known from the perturbation theory of multiple eigenvalues; see Davis and Kahan (1970), Friedland (1978) or Kato (1966). The dimension argument is essentially the same as the phenomenon known in quantum mechanics as the "crossing rule" of von Neuman and Wigner (1929); see also Friedland, Robbin and Sylvester (1984).…”

Canonical Forms of 3 x 3 Strongly and Nonstrictly Hyperbolic Systems with Complex Constant Coefficients

“…However, we rename them for the present section to look natural as a classification with a real constant a satisfying 0 < a < 1, or to a hermitian family. where the real constants a, jS, y and 3 satisfy 0 < a < 1 , 7 >-(ft 2 …”

“…. , n) such that And we denote this equivalence relation by By using the above operations a) and b), it is easy to see that any matrix family is equivalent to some <5 l5 ..., B n > where B l9 B 2) ..., B n are linearly independent and none of their nonzero linear combinations is equal to a scalar multiple of identity. Let us define a word indicating this property for later convenience.…”