QR Decomposition and Full Column Rank of A

In summary, A matrix is nonsingular if and only if it is of full column rank in a QR decomposition, and a matrix is singular if it has an eigenvalue that is zero. In a QR decomposition, the only way for the matrix R to be singular is if one or more of its diagonal values is zero, which would happen if the Householder matrix at some step transforms the current column into the zero vector instead of a constant times e1. Therefore, to prove that R is nonsingular, we need to show that all of its diagonal values are non-zero.
  • #1
gucci1
13
0
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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  • #2
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.
 

FAQ: QR Decomposition and Full Column Rank of A

What is QR Decomposition?

QR Decomposition is a mathematical method used to decompose a matrix into an orthogonal matrix and an upper triangular matrix. This method is commonly used in linear algebra and numerical analysis for solving systems of linear equations and finding the eigenvalues of a matrix.

How is QR Decomposition performed?

QR Decomposition is performed by using the Gram-Schmidt process, which involves orthogonalizing the columns of a matrix to form an orthogonal matrix. The upper triangular matrix is then obtained by applying the orthogonal matrix to the original matrix.

What is the significance of QR Decomposition?

QR Decomposition is important because it can simplify many problems in linear algebra, such as solving systems of linear equations, finding the inverse of a matrix, and computing eigenvalues and eigenvectors. It also provides a more stable and efficient method for solving these problems compared to other methods.

What is the Full Column Rank of a matrix?

The Full Column Rank of a matrix is the maximum number of linearly independent columns in the matrix. This means that the columns of the matrix can't be written as a linear combination of other columns. A matrix with full column rank is said to be full rank, and it has a unique inverse.

How is the Full Column Rank of a matrix related to QR Decomposition?

The Full Column Rank of a matrix is closely related to QR Decomposition because if a matrix has full column rank, then it can be decomposed into an orthogonal matrix and an upper triangular matrix using the QR Decomposition method. Additionally, the rank of the upper triangular matrix obtained from QR Decomposition will be equal to the Full Column Rank of the original matrix.

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