  Braid matrices $\hat{R}(\theta)$, corresponding to vector representations,
are spectrally decomposed obtaining a ratio $f_{i}(\theta)/f_{i}(-\theta)$ for
the coefficient of each projector $P_{i}$ appearing in the decomposition. This
directly yields a factorization $(F(-\theta))^{-1}F(\theta)$ for the braid
matrix, implying also the relation $\hat{R}(-\theta)\hat{R}(\theta)=I$.This is
achieved for $GL_{q}(n),SO_{q}(2n+1),SO_{q}(2n),Sp_{q}(2n)$ for all $n$ and
also for various other interesting cases including the 8-vertex matrix.We
explain how the limits $\theta \to \pm \infty$ can be interpreted to provide
factorizations of the standard (non-Baxterized) braid matrices. A systematic
approach to diagonalization of projectors and hence of braid matrices is
presented with explicit constructions for
$GL_{q}(2),GL_{q}(3),SO_{q}(3),SO_{q}(4),Sp_{q}(4)$ and various other cases
such as the 8-vertex one. For a specific nested sequence of projectors
diagonalization is obtained for all dimensions. In each factor $F(\theta)$ our
diagonalization again factors out all dependence on the spectral parameter
$\theta$ as a diagonal matrix. The canonical property implemented in the
diagonalizers is mutual orthogonality of the rows. Applications of our
formalism to the construction of $L-$operators and transfer matrices are
indicated. In an Appendix our type of factorization is compared to another one
proposed by other authors.
