Triangular decomposition of a Gram matrix

The triangular decomposition of a Gram matrix as a method for finding the QR decomposition of a square matrix [math]A[/math] works only if the non-singularity of the original matrix is guaranteed. The method consists of three parts: 1. Construction of the Gram matrix [math]A^*A[/math] for the columns of the original matrix. 2. Finding the Cholesky decomposition [math]R^*R[/math] of the Gram matrix [math]A^*A[/math]. 3. Calculation of the unitary matrix [math]Q=AR^{-1}[/math] by using, for instance, the modified back substitution.

The method is not used in practice not only because of the restriction caused by the non-singularity requirement but also for the simple reason that the condition number of the Gram matrix is the square of the condition number of the original matrix.