QR Decomposition and Full Column Rank of A

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SUMMARY

The discussion centers on proving that the upper triangular matrix R in a QR decomposition is nonsingular if and only if the matrix A has full column rank. Participants highlight the significance of the diagonal elements of R, noting that R is singular when any diagonal element is zero. The conversation emphasizes the relationship between the Householder transformations and the potential for a column of A to be transformed into the zero vector, which would indicate a loss of rank.

PREREQUISITES
  • Understanding of QR decomposition methods, specifically Gram-Schmidt and Householder transformations.
  • Knowledge of matrix rank and the concept of full column rank.
  • Familiarity with properties of triangular matrices and eigenvalues.
  • Basic linear algebra concepts, including matrix singularity and diagonalization.
NEXT STEPS
  • Study the properties of Householder transformations in detail.
  • Learn about the implications of matrix rank in linear algebra.
  • Explore the relationship between eigenvalues and matrix singularity.
  • Review proofs related to QR decomposition and matrix rank to solidify understanding.
USEFUL FOR

Students and professionals in mathematics, particularly those studying linear algebra, as well as researchers working with matrix decompositions and their applications in numerical analysis.

gucci1
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Hey guys,

I have a problem where I am supposed to prove that R is nonsingular iff A is of full column rank in a QR decomposition.

I feel like I fully understand the two major processes for obtaining a QR decomposition (Gram-Schimdt and Householder Transformations), however, I am not entirely sure how to prove this problem.

I know that the only way that R is singular is if one or more of its diagonal values is zero, and this would only happen if the Householder matrix at some step transforms the current column into the zero vector instead of a constant times e1 (the first column of the identity).

Does anyone have suggestions for how to start proving this? I really don't even know what my first step is :-/ Thanks for any help you can offer,

gucci
 
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gucci said:
Hey guys,

I have a problem where I am supposed to prove that R is nonsingular iff A is of full column rank in a QR decomposition.

I feel like I fully understand the two major processes for obtaining a QR decomposition (Gram-Schimdt and Householder Transformations), however, I am not entirely sure how to prove this problem.

I know that the only way that R is singular is if one or more of its diagonal values is zero, and this would only happen if the Householder matrix at some step transforms the current column into the zero vector instead of a constant times e1 (the first column of the identity).

Does anyone have suggestions for how to start proving this? I really don't even know what my first step is :-/ Thanks for any help you can offer,

gucci

Hi gucci!

A matrix is singular iff it has an eigenvalue that is zero.

And a triangular matrix has its eigenvalues is on its diagonal.
 

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