We present a new numerical method for computing selected eigenvalues and eigenvectors of the two-parameter eigenvalue problem. The method does not require good initial approximations and is able to tackle large problems that are too expensive for methods that compute all eigenvalues. The new method uses a two-sided approach and is a generalization of the Jacobi-Davidson type method for right definite two-parameter eigenvalue problems [M. E. Hochstenbach and B. Plestenjak, SIAM J. Matrix Anal. Appl., 24 (2002), pp. 392-410]. Here we consider the much wider class of nonsingular problems. In each step we first compute Petrov triples of a small projected two-parameter eigenvalue problem and then expand the left and right search spaces using approximate solutions to appropriate correction equations. Using a selection technique, it is possible to compute more than one eigenpair. Some numerical examples are presented.
Abstract. Let B be a nilpotent matrix and suppose that its Jordan canonical form is determined by a partition λ. Then it is known that its nilpotent commutator NB is an irreducible variety and that there is a unique partition µ such that the intersection of the orbit of nilpotent matrices corresponding to µ with NB is dense in NB. We prove that map D given by D(λ) = µ is an idempotent map. This answers a question of Basili and Iarrobino [9] and gives a partial answer to a question of Panyushev [18]. In the proof, we use the fact that for a generic matrix A ∈ NB the algebra generated by A and B is a Gorenstein algebra. Thus, a generic pair of commuting nilpotent matrices generates a Gorenstein algebra. We also describe D(λ) in terms of λ if D(λ) has at most two parts.
Abstract. For every smooth (irreducible) cubic surface S we give an explicit construction of a representative for each of the 72 equivalence classes of determinantal representations. Equivalence classes (under GL3 × GL3 action by left and right multiplication) of determinantal representations are in one to one correspondence with the sets of six mutually skew lines on S and with the 72 (two-dimensional) linear systems of twisted cubic curves on S. Moreover, if a determinantal representation M corresponds to lines (a1, . . . , a6) then its transpose M t corresponds to lines (b1, . . . , b6) which together form a Schläfli's double-six`a 1 ...a 6 b 1 ...b 6´. We also discuss the existence of self-adjoint and definite determinantal representation for smooth real cubic surfaces. The number of these representations depends on the Segre type Fi. We show that a surface of type Fi, i = 1, 2, 3, 4 has exactly 2(i − 1) nonequivalent self-adjoint determinantal representations none of which is definite, while a surface of type F5 has 24 nonequivalent self-adjoint determinantal representations, 16 of which are definite.
We study components and dimensions of higher-order determinantal varieties obtained by considering generic m × n (m 6 n) matrices over rings of the form F[t]=(t k ), and for some ÿxed r, setting the coe cients of powers of t of all r × r minors to zero. These varieties can be interpreted as spaces of (k − 1)th order jets over the classical determinantal varieties; a special case of these varieties ÿrst appeared in a problem in commuting matrices. We show that when r = m, the varieties are irreducible, but when r ¡ m, these varieties are reducible. We show that when r = 2 ¡ m (any k), there are exactly k=2 + 1 components, which we determine explicitly, and for general r ¡ m, we show there are at least k=2 + 1 components. We also determine the components explicitly for k = 2 and 3 for all values of r (for k = 3 for all but ÿnitely many pairs of (m; n)).
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