# Evaluation of the square root of a complex matrix

• posted

Summary: How do I evaluate the square root of a complex matrix?

I am studying from Pieter LD Abrie's "Design of RF and Microwave Amplifiers and Oscillators", 2ed, Artech House, 2009.

In the first chapter he presents derivations of S-parameters for N-port net works (sec. 1.5), he states an expression on page 13, (sqrt(2))^(-1)*[Z_0 + Z_0*]^(1/2), where the matrix of impedances is Z_0 = [R_0i + jX­­_0i], and Z_0* is the complex conjugate matrix of Z_0; j=sqrt(-1) ; R_0i is the ith resistance of port i and X_0i is the ith reactance of por t i.

I'm trying to derive this expression, but all the texts on Linear Algebra t hat I have in my personal library only deal with positive integer powers of real matrices. (I have in my arsenal "Elementary Linear Algebra: Applicat ions Version", H.Anton & C.Rorres, 6ed, Wiley: "Schaum's Outline of Linear Algebra; "Mathematical Methods for Physics and Engineering", 2ed, K.F.Riley , M.P.Hobson & S.J.Bence, Cambridge UP).

What topics and texts do I need to study to be able to evaluate the square root of a complex matrix?

Cheers, Julian

PS Anyone know of a forum where they provide/allow mathematical notation?

• posted

I'll take on the easy, last question. Perhaps upload a pdf of the equation(s) to DropBox and provide a secure link, e.g.,

That lets you use the full power of LaTeX to compose it, notes and all. Or, for a quick'n'dirty equation, EqualX uses LaTeX math mode commands and you can \mbox{} a short note if required.

• posted

and maybe...

or...

Looks like it requires "recursive" approaches... i.e. guessing solutions ;-) ...Jim Thompson

```--
| James E.Thompson                                 |    mens     |
• posted

Have a look at:

Bell Systems Technical Journal, (1941), Rice, S.O.,v.20,p.131-178. for a matrix solution of transmission lines.

I wrote a paper in 1950 on an extension to n-wire lines, in which the square root of a complex matrix was involved. In this case the root is unique, but in general it may not be.

```--
Virg Wall```
• posted

I think the only book I have that goes into matrix square roots is Dan Simon's book on Kalman filtering.

You might ask on sci.electronics.design or comp.dsp -- this isn't a basic question!!

Note that this is a USENET newsgroup, and is therefor text-only. Usually if there's a lot of math being flung around people will use LaTeX format, or pseudo-LaTeX. You can generally make yourself understood.

```--
Tim Wescott
Control system and signal processing consulting ```
• posted

If you just want to numerically evaluate a result, you may simply use octave (or Matlab)

octave:1> sqrt([1 j;2 j]) ans =

1.00000 + 0.00000i 0.70711 + 0.70711i 1.41421 + 0.00000i 0.70711 + 0.70711i

If you want to derive the whole expression, I would investigate the meaning of the parameters instead of relying on pure algebra. For instance, the relation between Z and S parameters given in the wikipedia

Z=sqrt(z)*(1+S)(1-S)^(-1)*sqrt(z)

is just the vectorial generalization of the scalar equation

Z=Z0*(1+ro)/(1-ro)

I know that this is not the equation you are looking for, but the idea may be a starting point.

Pere

• posted

Thank you, gentlemen one and all, for your kind and helpful responses. I am evaluating your advice, as time permits, and hope to upload a pdf file of the excerpt from Abrie, 2009, sec 1.5 to Dropbox soon.

Kind regards,

Julian Grodzicky

ElectronDepot website is not affiliated with any of the manufacturers or service providers discussed here. All logos and trade names are the property of their respective owners.