2,405 research outputs found
Polynomial identities for hypermatrices
We develop a method to construct algebraic invariants for hypermatrices. We
then construct hyperdeterminants and exhibit a generalization of the
Cayley-Hamilton theorem for hypermatrices.Comment: 65 pages. Several results expanded. Title changed to better reflect
the content and agree with published preprint versio
On some algebraic identities and the exterior product of double forms
We use the exterior product of double forms to reformulate celebrated
classical results of linear algebra about matrices and bilinear forms namely
the Cayley-Hamilton theorem, Laplace expansion of the determinant, Newton
identities and Jacobi's formula for the determinant. This new formalism is then
used to naturally generalize the previous results to higher multilinear forms
namely to double forms.
In particular, we show that the Cayley-Hamilton theorem once applied to the
second fundamental form of a hypersurface of the Euclidean space is equivalent
to a linearized version of the Gauss-Bonnet theorem, and once its
generalization is applied to the Riemann curvature tensor (seen as a
double form) is an infinitisimal version of the general Gauss-Bonnet-Chern
theorem. In addition to that, the general Cayley-Hamilton theorems generate
several universal curvature identities. The generalization of the classical
Laplace expansion of the determinant to double forms is shown to lead to new
general Avez type formulas for all Gauss-Bonnet curvatures.Comment: 32 pages, in this new version we added: an introduction to the
exterior and composition products of double forms, a new section about
hyperdeterminants and hyperpfaffians and reference
Invariant and polynomial identities for higher rank matrices
We exhibit explicit expressions, in terms of components, of discriminants,
determinants, characteristic polynomials and polynomial identities for matrices
of higher rank. We define permutation tensors and in term of them we construct
discriminants and the determinant as the discriminant of order , where
is the dimension of the matrix. The characteristic polynomials and the
Cayley--Hamilton theorem for higher rank matrices are obtained there from
The Exponential Map for the Conformal Group 0(2,4)
We present a general method to obtain a closed, finite formula for the
exponential map from the Lie algebra to the Lie group, for the defining
representation of the orthogonal groups. Our method is based on the
Hamilton-Cayley theorem and some special properties of the generators of the
orthogonal group, and is also independent of the metric. We present an explicit
formula for the exponential of generators of the groups, with , in particular we are dealing with the conformal group , which
is homomorphic to the group. This result is needed in the
generalization of U(1) gauge transformations to spin gauge transformations,
where the exponential plays an essential role. We also present some new
expressions for the coefficients of the secular equation of a matrix.Comment: 16pages,plain-TeX,(corrected TeX
On quantum matrix algebras satisfying the Cayley-Hamilton-Newton identities
The Cayley-Hamilton-Newton identities which generalize both the
characteristic identity and the Newton relations have been recently obtained
for the algebras of the RTT-type. We extend this result to a wider class of
algebras M(R,F) defined by a pair of compatible solutions of the Yang-Baxter
equation. This class includes the RTT-algebras as well as the Reflection
equation algebras
- …