  We present the realization of Hurwitz algebras in terms of 2x2 vector
matrices, which maintain the correspondence between the geometry of the vector
spaces used in the classical physics and the underlined algebraic foundation of
the quantum theory. The multiplication rule used is modification of the one
originally introduced by M.Zorn. We demonstrate that our multiplication is not
intrinsically non-associative; the realization of the real and complex numbers
is commutative and associative, the real quaternions maintain associativity and
the real octonion matrices form an alternative algebra. The extension to the
calculus of the matrices (with Hurwitz algebra valued matrix elements) of the
arbitrary dimensions is straightforward. We discuss briefly the applications of
the obtained results to the extensions of the standard Hilbert space
formulation of the quantum physics and to the alternative wave mechanical
formulation of the classical field theory.
