Discussion Overview
The discussion revolves around the rank of a matrix A with a reduced QR-decomposition, specifically focusing on the relationship between the number of nonzero diagonal elements of matrix R and the rank of matrix A. The scope includes theoretical aspects of linear algebra.
Discussion Character
Main Points Raised
- One participant argues that if matrix A has a reduced QR-decomposition with R having j nonzero diagonal elements, then the rank of A is exactly j.
- Another participant questions the precision of the claim regarding R having "j nonzero diagonal elements," suggesting it could also mean "at least j nonzero elements."
- A different participant supports the claim that A's rank should be exactly j, explaining that the remaining columns in Q associated with the zero-valued elements of R span A's null space.
- It is noted that rank(A) equals rank(QR) equals rank(R), with the rank of R being j.
- One participant points out that the statement about Q covering the full basis is imprecise, clarifying that Q is an invertible orthogonal (or unitary) matrix, which is sufficient for the argument.
Areas of Agreement / Disagreement
Participants express disagreement regarding whether the rank of A is at least j or exactly j, with some supporting the latter view while others raise questions about the precision of the definitions involved.
Contextual Notes
The discussion highlights potential ambiguities in the definitions of nonzero diagonal elements and the implications for the rank of matrix A. There are unresolved nuances regarding the interpretation of the QR-decomposition and its relationship to the rank.