# Complex Cholesky Decomposition

1. Jul 22, 2008

### 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.

"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.

2. Jul 22, 2008

### maze

3. Jul 22, 2008

### magda3227

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

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• ###### chol2.png
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4. Jul 22, 2008

### 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)

5. Jul 22, 2008

### magda3227

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

Thanks once again.