Generalized elementary symmetric functions and quaternion matrices

Generalized Symmetric Functions and Invariants of Matrices

It is well known that over an infinite field the ring of symmetric functions in a finite number of variables is isomorphic to the one of polynomial functions on a single matrix that are invariants by the action of conjugation by general linear group. We generalize this result showing that the abelianization of the algebra of the symmetric tensors of fixed order over a free associative algebra i...

On the elementary symmetric functions of a sum of matrices

Often in mathematics it is useful to summarize a multivariate phenomenon with a single number. In fact, the determinant – which is denoted by det – is one of the simplest cases and many of its properties are very well-known. For instance, the determinant is a multiplicative function, i.e. det(AB) = detA · detB, A,B ∈ Mn, and it is a multilinear function, but it is not, in general, an additive f...

A brief introduction to quaternion matrices and linear algebra and on bounded groups of quaternion matrices

The division algebra of real quaternions, as the only noncommutative normed division real algebra up to isomorphism of normed algebras, is of great importance. In this note, first we present a brief introduction to quaternion matrices and quaternion linear algebra. This, among other things, will help us present the counterpart of a theorem of Herman Auerbach in the setting of quaternions. More ...

Generalized Functions of Symmetric Matrices Marvin Marcus1 and Morris Newman

The exponential functions of central-symmetric $X$-form matrices

It is well known that the matrix exponential function has practical applications in engineering and applied sciences. In this paper, we present some new explicit identities to the exponential functions of a special class of matrices that are known as central-symmetric $X$-form. For instance, $e^{mathbf{A}t}$, $t^{mathbf{A}}$ and $a^{mathbf{A}t}$ will be evaluated by the new formulas in this par...