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 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?

 

[Rich Webb]

On Sat, 28 Sep 2013 05:38:18 -0700 (PDT), Julian Grodzicky
<grodzicky_j@yahoo.com.au> wrote:

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?

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.,
https://www.dropbox.com/s/ehiyi4nv97pepye/example.pdf
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.
http://equalx.sourceforge.net/

 

[Jim Thompson]

On Sat, 28 Sep 2013 05:38:18 -0700 (PDT), Julian Grodzicky
<grodzicky_j@yahoo.com.au> wrote:

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?

<http://tinyurl.com/mfhfx3g>

and maybe...

<http://www.mit.edu/~wingated/stuff_i_use/matrix_cookbook.pdf>

or...

<http://tinyurl.com/ny8nroe>

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

...Jim Thompson
--
| James E.Thompson | mens |
| Analog Innovations | et |
| Analog/Mixed-Signal ASIC's and Discrete Systems | manus |
| San Tan Valley, AZ 85142 Skype: Contacts Only | |
| Voice480)460-2350 Fax: Available upon request | Brass Rat |
| E-mail Icon at http://www.analog-innovations.com | 1962 |

I love to cook with wine. Sometimes I even put it in the food.

 

[VWWall]

Julian Grodzicky wrote:

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?

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

 

[Tim Wescott]

On Sat, 28 Sep 2013 05:38:18 -0700, Julian Grodzicky wrote:

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?

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

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

 

[o pere o]

On 09/28/2013 02:38 PM, Julian Grodzicky wrote:

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]

On Saturday, September 28, 2013 10:38:18 PM UTC+10, Julian Grodzicky wrote:

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