Evaluation of the square root of a complex matrix

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Julian Grodzicky 12 years ago

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?

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Rich Webb 12 years ago

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.,

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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.

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Jim Thompson 12 years ago

and maybe...

or...

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

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VWWall 12 years ago

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

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Tim Wescott 12 years ago

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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 www.wescottdesign.com

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o pere o 12 years ago

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

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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

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Julian Grodzicky 12 years ago

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

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Evaluation of the square root of a complex matrix

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