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QR Decomposition and Full Column Rank of A

gucci

New member
Sep 1, 2013
13
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
 

Klaas van Aarsen

MHB Seeker
Staff member
Mar 5, 2012
8,791
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.