Signatures of quadratic forms over formally real fields have been generalized in [BP2] to hermitian forms over central simple algebras with involution over such fields. This was achieved by means of an application of Morita theory and a reduction to the quadratic form case. A priori, signatures of hermitian forms can only be defined up to sign, i.e., a canonical definition of signature is not p...

Let M be an n-dimensional projective algebraic manifold in certain projective space CP . The hyperplane line bundle of CP restricts to an ample line bundle L on M , which is called a polarization of M . A Kähler metric g is called a polarized metric, if the corresponding Kähler form represents the first Chern class c1(L) of L in H (M,Z). Given any polarized Kähler metric g, there is a Hermitian...

The inertia of a Hermitian matrix is defined to be a triplet composed by the numbers of the positive, negative and zero eigenvalues of the matrix counted with multiplicities, respectively. In this paper, we give various closed-form formulas for the maximal and minimal values for the rank and inertia of the Hermitian expression A + X, where A is a given Hermitian matrix and X is a variable Hermi...

The QR method is a very eecient method for computing the spectrum of Hermitian tridiagonal matrices since the tridiagonal form is preserved over the iterations. For non-Hermitian tridiagonal matrices the QR method destroys the tridiagonal form. In this report we study two methods, the HR and the XHR methods, that preserve tridiagonal form for pseudo-Hermitian matrices. We also report results fr...

KT-geometry is the geometry of a Hermitian connection whose torsion is a 3-form. HKT-geometry is the geometry of a hyper-Hermitian connection whose torsion is a 3-form. We identify non-trivial conditions for a reduction theory for these types of geometry.

We study projective homogeneous varieties under an action of a projective unitary group (of outer type). We are especially interested in the case of (unitary) grassmannians of totally isotropic subspaces of a hermitian form over a field, the main result saying that these grassmannians are 2-incompressible if the hermitian form is generic. Applications to orthogonal grassmannians are provided.

Coherent states (CS) for non-Hermitian systems are introduced as eigenstates of pseudo-Hermitian boson annihilation operators. The set of these CS includes two subsets which form bi-normalized and bi-overcomplete system of states. The subsets consist of eigenstates of two complementary lowering pseudo-Hermitian boson operators. Explicit constructions are provided on the example of one-parameter...

For an Hermitian matrix the QR transform is diagonally similar to two steps of the LR transforms. Even for non-Hermitian matrices the QR transform may be written in rational form.

Let K be a field (characteristic # 2), and let L be a quadratic extension of K. Then L = K{yja) for some aeK. Let D be the quaternion algebra (a, b/K) generated by elements i, j satisfying i = a, j 2 = b, ij = —ji. We will assume that D is a division algebra, i.e. that the quadratic form <1> ~> ~b, ab} is anisotropic. Let — denote the standard involution on L and on D so that y/a = —*Ja on L an...