What are the properties of matrix inversion using the Cholesky decomposition for non symmetric matrices?

Question

I managed to find some information on the properties of matrix inversion using the LU decomposition. There is also some material on doing the inversion using the Cholesky decomposition for non-symmetric matrices.

I haven't been able to find any information on how Cholesky decomposition inversion behaves in practice. Would it be on par with the LU decomposition inversion in terms of speed and stability?

Answer 1

Inverting a nonsymmetric matrix $A$ by $A^{-1}=A^T(AA^T)^{-1}$ using the Cholesky factorization of $AA^T$ is a pretty bad idea both in terms of complexity and numerical stability.

There is absolutely nothing saved in terms of the number of operations. Forming the product $AA^T$ prior to its factorization involves about $n^3$ flops (yes, yes, Strassen gives a better exponent). With the cost of $n^3/3$ flops of the Cholesky factorization this gives $4n^3/3$ flops in total. This is twice more expensive than LU and just same as the Householder QR factorization for $A$ directly.

Not only this approach of inversion is not faster in any way, it is also inferior to almost any other method one might think of. The numerical errors committed by solving systems like $AA^T$ are proportional to the square of the condition number of $A$ which in turn roughly halves the number of valid digits in the solution compared to more stable methods based on LU or QR.

Answer 2

It is much easier to compute the inverse of a triangular matrix and there exist numerical solutions. Then the original matrix inverse is computed simply by multiplying the two inverses as $$ A^{-1} = \left(L^{-1}\right)^T L^{-1} $$ for $L$ being lower-triangular from Cholesky decomposition $A = LL^T.$

EDIT: For more optimized version see here.

Answer 3

There is no such thing as a Cholesky decomposition for non symmetric matrices.

As the decomposition has the form

$$A=LL^T$$ you obviously must have

$$A^T=A.$$