Using the idea of the generating function of a matrix in an extended sense we establish a Bezoutian type formula for a matrix satisfying an intertwining relation of the form . In the particular case of classical generating functions this formula gives a simple proof of Lander’s theorem on the inverse of a Hankel matrix.
The purpose of the Part I of this paper is to develop the geometry of Gram's determinants in Hilbert space. In Parts II and III a generalization is given of the Pythagorean theorem and triangular inequality for finite vector families.
The article deals with bundles of linear algebra as a specifications of the case of smooth manifold. It allows to introduce on smooth manifold a metric by a natural way. The transfer of geometric structure arising in the linear spaces of associative algebras to a smooth manifold is also presented.
We study some geometric properties associated with the -geometric means of two positive definite matrices and . Some geodesical convexity results with respect to the Riemannian structure of the positive definite matrices are obtained. Several norm inequalities with geometric mean are obtained. In particular, we generalize a recent result of Audenaert (2015). Numerical counterexamples are given for some inequality questions. A conjecture on the geometric mean inequality regarding pairs...
The article deals with spaces the geometry of which is defined by cyclic and anticyclic algebras. Arbitrary multiplicative function is taken as a fundamental form. Motions are given as linear transformation preserving given multiplicative function.
A real matrix is a G-matrix if is nonsingular and there exist nonsingular diagonal matrices and such that , where denotes the transpose of the inverse of . Denote by a diagonal (signature) matrix, each of whose diagonal entries is or . A nonsingular real matrix is called -orthogonal if . Many connections are established between these matrices. In particular, a matrix is a G-matrix if and only if is diagonally (with positive diagonals) equivalent to a column permutation of...