Method level

Triangular decomposition of a Gram matrix

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