Matrix decomposition: Difference between revisions

Revision as of 18:30, 31 January 2021

Revision as of 13:45, 14 February 2021

Line 47:

*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 alternativeeffective way ofto solvingsolve 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^{\mathrm{T}}Q=I</math>, so that <math>Ax=b</math> is equivalent to <math>Rx=Q^{\mathrm{T}}b</math>, which is easiervery toeasy solve since <math>R</math> is [[triangular matrix|triangular]].

=== RRQR factorization ===