simplifying mathematics of symbolic expression

I have a symbolic matrix

[                 cos(conj(x))*cos(x),                 cos(conj(x))*exp(a*i)*sin(x)]
[ sin(conj(x))*exp(-conj(a)*i)*cos(x), sin(conj(x))*exp(a*i)*exp(-conj(a)*i)*sin(x)]

It looks so difficult and ugly. I am looking for simplified expression for the above something like the one below-

[       (cos(x))^2               cos(x)*exp(a*i)*sin(x)]
[sin(x)*exp(-conj(a)*i)*cos(x)         (sin(x))^2      ]

since x is angle it does not have conjugate so I would like to have cos(conj(x)) as cos(x) and therefore cos(conj(x))*cos(x) as (cos(x))^2 or even better cos2x where 2 is in superscript as we normally have while writing.

Also exp(a*i)*exp(-conj(a)*i) would normally have cancelled out to give 1 but the resulted expression doesn't simplify to it

Can anybody help me here

thanks very much

Answer 1

[Star Strider]

You are not telling the Symbolic Toolbox everything you know about your system. See if this code does something similar to what you want:

syms x a 
assume(a, 'real')
assumeAlso(x, 'real')

f = [ [ cos(conj(x))*cos(x), cos(conj(x))*exp(a*i)*sin(x)]; [ sin(conj(x))*exp(-conj(a)*i)*cos(x), sin(conj(x))*exp(a*i)*exp(-conj(a)*i)*sin(x)] ]

fs = simplify(collect(f))

Note that when it knows that x is real, it eliminates conj(x).

‘Also exp(a*i)*exp(-conj(a)*i) would normally have cancelled out to give 1 but the resulted expression doesn't simplify to it’

It does now that it knows what you know about a:

g = exp(a*i)*exp(-conj(a)*i)

Experiment with it and the various functions in the Symbolic Toolbox to get the result in the form you want.

If you want it as a function you can execute as regular MATLAB code (outside of the Symbolic Math Toolbox), see matlabFunction.

[Suma]

Thank you,

I have a minor mistake while writing my problem above and want to ask this before I close this thread.

Lets say x is complex.

If x is a complex number then x*conj(x) is x^2 (modulus square), how do I achieve x^2 form if x is complex instead of real?

Thanks

[Walter Roberson]

That is not correct.

(3+5*i)^2 = -16+30*i
(3+5*i)*conj(3+5*i) = 34
abs(3+5*i) = sqrt(34)

so x*conj(x) = abs(x)^2 rather than x^2

[Suma]

Hi,

I want to have x*conj(x) reduced to abs(x)^2 in expression, how can I do that?

[Star Strider]

The only way I can think of to do that is to create your own anonymous function:

syms rex imx 'real'

abs2 = @(x) simplify(x*conj(x));

x = rex + j*imx;

b = abs2(x)

produces:

b =

imx^2 + rex^2

Otherwise, you will have to specify what you want as abs(x)^2 in your code.

[Walter Roberson]

What result do you get from

evalin(symengine, 'simplify(x*conjugate(x)) assuming (x, Type::Complex)')

[Suma]

Hello,

@star method i don't understand and probably not the thing I was looking for.

@walter, the code provided didn't produce a condensed form for x*conj(x), also I wish that the trick also did produce cos(conj(x))*cos(x) as (cos(x))^2

[Walter Roberson]

To check you used "conjugate" in the test I asked you to try, and not "conj" ? The code I asked to test needs conjugate spelt out in full.

[Suma]

I used conjugate as you told

[Star Strider]

The Wolfram online calculator gives a number of different results, none of which is what you specified. See http://www.wolframalpha.com/input/?i=cos%28conj%28r%2Bi*q%29%29*cos%28r%2Bi*q%29+ for details. (I used r for the real part and q for the imaginary.)

[Walter Roberson]

Try with x = 2 + 3*i and you will see that cos(conj(x))*cos(x) is not the same as (cos(x))^2

[Suma]

ok thanks