# Biorthogonality

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. ${\ displaystyle Q_ {k} \ in \ mathbb {C} ^ {n, k}}$${\ displaystyle {\ hat {Q}} _ {k} \ in \ mathbb {C} ^ {n, k}}$${\ displaystyle {\ hat {Q}} _ {k} ^ {H} Q_ {k} = D_ {k}}$${\ displaystyle D_ {k}}$

The matrices are biorthonormal if the diagonal matrix is ​​the identity , i.e. if . The definitions for orthogonality and orthonormality are obtained by choosing. ${\ displaystyle {\ hat {Q}} _ {k} ^ {H} Q_ {k} = I_ {k}}$${\ displaystyle {\ hat {Q}} _ {k} = Q_ {k}}$

Biorthogonality occurs in the context of the asymmetrical Lanczos method and the two-sided Gram-Schmidt .