Does Cholesky Decomposition Change for Complex Matrices?

[magda3227]

I am having trouble finding information about decomposing a complex symmetric positive definite matrix. I was wondering if the cholesky decomposition would change to accommodate complex numbers. I understand that multiplying, dividing, and taking the square root of complex numbers is not the same as with real numbers, but would any additional components need to be added to the algorithm?

I only ask because I am writing a C program and need to implement the Cholesky function. I have written code that works properly for the real decomposition, but when I try to use complex numbers, I am not getting the correct answer.

I did read this, however...
"All the square roots appearing are real. So if one writes a computer programme
implementing the Cholesky factorisation one can be sure that no
complex numbers appear in the course of the computation."

How would this be so? The answer has imaginary parts, not on the diagonals (is that what this means), but elsewhere.

 

[maze]

Yeah, you just take the complex conjugate transpose instead of the regular transpose
M = L L*

* means complex conjugate transpose

http://en.wikipedia.org/wiki/Cholesky_decomposition

 

[magda3227]

So how would that be incorporated into the equations in the attachments...

 

[maze]

You would use the equations in the attachment to find L. To find L*, you take the transpose of L, and then wherever there is something like 3-2i, change it to 3+2i (5+.9i -> 5-.9i, etc)

 

[magda3227]

Thank you for you help. I was truly over complicating it.

Thanks once again.