Difference between revisions of "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 <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 [[ | + | 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 method)|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. | 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. | ||
Revision as of 17:02, 16 March 2018
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.
