  The Gauss-Borel or $LU$ factorization of Gram matrices of bilinear forms is
the pivotal element in the discussion of the theory of biorthogonal
polynomials. The construction of biorthogonal families of polynomials and its
second kind functions, of the spectral matrices modeling the multiplication by
the independent variable $x$, the Christoffel-Darboux kernel and its projection
properties, are discussed from this point of view. Then, the Hankel case is
presented and different properties, specific of this case, as the three terms
relations, Heine formulas, Gauss quadrature and the Christoffel-Darboux formula
are given. The classical orthogonal polynomial of Hermite, Laguerre and Jacobi
type are discussed and characterized within this scheme. Finally, it is shown
who this approach is instrumental in the derivation of Christoffel formulas for
general Christoffel and Geronimus perturbations of the bilinear forms.
