  The group $SL(2,\mathbb{C})$ of all complex $2\times 2$ matrices with
determinant one is closely related to the group
$\boldsymbol{\mathcal{L}}_{+}^\uparrow$ of real $4\times 4$ matrices
representing the restricted Lorentz transformations. This relation, sometimes
called the spinor map, is of fundamental importance in relativistic quantum
mechanics and has applications also in general relativity. In this paper we
show how the spinor map may be expressed in terms of pure matrix algebra by
including the Kronecker product between matrices in the formalism. The
so-obtained formula for the spinor map may be manipulated by matrix algebra and
used in the study of Lorentz transformations.
