Let a positive definite matrix(adsbygoogle = window.adsbygoogle || []).push({}); Abe factorized toPandQ,A=P*Qand let an arbitrary matrixB.

I am calculating the relative error of the factorization through the norm:

[itex]\epsilon = \left\| \textbf{A}-\textbf{PQ} \right\| / \left\| \textbf{A} \right\|[/itex]

which gives

[itex]\epsilon <1\text{e}-16[/itex]

so I assume factorization is correct.

But things go messy when I try to multiply the factorized form of A with B.

In particular, the relative error,r, of the product

[itex]r = \left\| \textbf{AB}-\textbf{PQB} \right\| / \left\| \textbf{AB} \right\|[/itex]

now bloats, i.e. I get

[itex]r>0.1.[/itex]

Note thatBis arbitrary, in particular I have tried several different types: random, structured, all-ones matrix, even the identity matrix.

I'm confused. How come factorization is correct and then the multiplication bloats?

Has anything to do with condition number?

(Unfortunately I can't disclose the type of factorization but I can tell that P and Q are not triangular)

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# Multiplication bloards after factorization

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