Matrix decomposition: Difference between revisions

Revision as of 05:25, 18 December 2020

Revision as of 11:38, 12 January 2021

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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-''m'', and ''<math>R''</math> is an [[triangular matrix|upper triangular]] matrix of size ''m''-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 alternative way of solving the system of equations <math>Ax=b</math> without [[matrix inverse|inverting]] the matrix ''<math>A''</math>. The fact that ''<math>Q''</math> is [[orthogonal matrix|orthogonal]] means that <math>Q^TQ{\mathrm{T}}Q=I</math>, so that <math>Ax=b</math> is equivalent to <math>Rx=Q^Tb{\mathrm{T}}b</math>, which is easier to solve since ''<math>R''</math> is [[triangular matrix|triangular]].

=== RRQR factorization ===