Biorthogonality is a modification of the well-known orthogonality . One speaks of biorthogonal matrices and , if the column vectors are perpendicular to one another , where denotes a diagonal matrix.
The matrices are biorthonormal if the diagonal matrix is the identity , i.e. if . The definitions for orthogonality and orthonormality are obtained by choosing.
Biorthogonality occurs in the context of the asymmetrical Lanczos method and the two-sided Gram-Schmidt .