This paper is concerned with the generalized spectral decomposition or eigenprojector form of a linear operator over any field that is a splitting field of its minimal polynomial. A number system is constructed which is isomorphic to the factor ring \(\mathbb{C} [\lambda]/ \langle\psi \rangle\) for an arbitrary polynomial \(\psi\). If \(\psi\) is the minimal polynomial of a given linear operator, then its eigenprojector form is immediately determined. A novel proof is given based on the algebraic properties of idempotents and nilpotents. The eigenvector form extended to any field by \textit{L. E. Dickson} [Am. J. Math. 24, 101-108 (1902; JFM 33.0151.01)] can be applied to a linear operator over a finite field, providing it is the splitting field of the minimal polynomial. A Clifford algebra arises when a grading onto the algebra of endomorphisms on a finite-dimensional vector space is introduced together with a Hermitian conjugation. This is done most naturally in the case of \(\text{End} (\mathbb{C}^{2^n})\) by constructing a simple isomorphism to the corresponding Clifford algebra. The consequences of this isomorphism for Hermitian operators are given and a new polar form for Clifford numbers is derived, such as asserted in the following theorem: Every \({\mathfrak a} \in C1_{k,k+1}\) can be represented in the form \({\mathfrak a} = h_1u=h_1 \exp (jh_2)\), where \(h_1\) and \(h_2=-j\ln u\) are Hermitian.