Evaluation of the square root of a complex matrix

[Julian Grodzicky]

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 Amplifiersand Oscillators", 2ed, Artech House, 2009.

In the first chapter he presents derivations of S-parameters for N-port networks (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 port i.

I'm trying to derive this expression, but all the texts on Linear Algebra that I have in my personal library only deal with positive integer powers ofreal matrices. (I have in my arsenal "Elementary Linear Algebra: Applications 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?

 

[o pere o]

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 networks (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 port i.

I'm trying to derive this expression, but all the texts on Linear
Algebra that I have in my personal library only deal with positive
integer powers of real matrices. (I have in my arsenal "Elementary
Linear Algebra: Applications 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?

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
(http://en.wikipedia.org/wiki/Impedance_parameters),

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

 

[Julian Grodzicky]

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

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