Matrix decomposition: Difference between revisions

Revision as of 22:58, 18 August 2023

Revision as of 04:24, 28 November 2023

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{{main|QR decomposition}}

*Applicable to: ''m''-by-''n'' matrix ''A'' with linearly independent columns

*Decomposition: <math>A=QR</math> where <math>Q</math> is a [[unitary matrix]] of size ''m''-by-''mn'', and <math>R</math> is an [[triangular matrix|upper triangular]] matrix of size ''mn''-by-''n''

*Uniqueness: In general it is not unique, but if <math>A</math> is of full [[Matrix rank|rank]], then there exists a single <math>R</math> that has all positive diagonal elements. If <math>A</math> is square, also <math>Q</math> is unique.

*Comment: The QR decomposition provides an effective way to solve the system of equations <math>A \mathbf{x} = \mathbf{b}</math>. The fact that <math>Q</math> is [[orthogonal matrix|orthogonal]] means that <math>Q^{\mathrm{T}}Q=I</math>, so that <math>A \mathbf{x} = \mathbf{b}</math> is equivalent to <math>R \mathbf{x} = Q^{\mathsf{T}} \mathbf{b}</math>, which is very easy to solve since <math>R</math> is [[triangular matrix|triangular]].