Matrix decomposition: Difference between revisions

Revision as of 20:09, 19 May 2022

Revision as of 11:04, 20 August 2022

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{{main|Interpolative decomposition}}

=== Bidiagonal decomposition ===

*Applicable to: (n+1)-by-(n+1) matrix ''A''

*Decomposition:A = L1.L2...Ln.D.Un...U2.U1, where every <math>L</math> is lower bidiagonal, <math>D</math> is diagonal, and every <math>U</math> is upper bidiagonal. It is performed using Neville elimination.

*Comment: It is used for accurate computations, such as inverse matrix, eigenvalues, and zero Jordan block computation, and linear system solving, with totally nonnegative matrices.

*Uniqueness: It is unique under the condition that all L and U matrices are unit bidiagonal. However, new algorithms have dropped this requirement, making computations with singular matrices possible.

== Decompositions based on eigenvalues and related concepts ==

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*{{citation|last=Townsend|first=A.|last2=Trefethen|first2=L. N.|year=2015|title=Continuous analogues of matrix factorizations|journal=[[Proceedings of the Royal Society|Proc. R. Soc. A]]|volume=471|issue=2173|pages=20140585|doi=10.1098/rspa.2014.0585|pmid=25568618|pmc=4277194|bibcode=2014RSPSA.47140585T}}

*{{citation|last=Jun|first=Lu|year=2021|title=Numerical matrix decomposition and its modern applications: A rigorous first course|url=https://arxiv.org/abs/2107.02579|arxiv=2107.02579|access-date=2021-11-17}}