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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 [[Метод_Холецкого_(нахождение_симметричного_треугольного_разложения)|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.
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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 [[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.