  The interpretation of quantum mechanics due to Lande' is applied to the
connection between wave mechanics and matrix mechanics. The connection between
the differential eigenvalue equation and the matrix eigenvalue equation for an
operator is elucidated. In particular, we show that the elements of a matrix
vector state are probability amplitudes with a structure rather than being mere
constants. We obtain the most general expressions for the probability
amplitudes for the description of spin-1/2 measurements. As a result, we derive
spin-1/2 operators and vectors from first principles. The procedure used is
analogous to that by which orbital angular momentum wavefunctions and operators
are transformed to matrix mechanics vectors and matrices. The most generalized
forms of the spin operators and their eigenvectors for spin-1/2 are derived and
shown to reduce to the Pauli spin matrices and vectors in an appropriate limit.
