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What dose this mean?

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Aisha Mohamed
Aisha Mohamed 2022년 8월 20일
댓글: Aisha Mohamed 2022년 8월 21일
I know the magnitude of complex number a+ib is the sqrt(a^2+b^2).and the magnitude of the complex function for examble (f(z)=ab/cd) where a,b,c and d are complex numbers is |f(z)|=|a||b|/|c||d|.
My question is:
I received this explenation from some expert in matlab work , but I can not understand it
((The magnitude of f(x) corresponds to rotating each point in the complex plane over to the positive x axes, preserving vector magnitude. The result has no remaining phase.))
Can I get more explenation or referece to understand this please?
I will appreciate any help

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Torsten
Torsten 2022년 8월 20일
The easiest way to see what you are asking is to use the polar representation of a complex number.
If z = r*exp(i*phi), then r is the magnitude of z and phi is the phase angle.
Rotation of a complex number w by an angle theta in the complex plane is given by multiplication of w by exp(i*theta):
w' = exp(i*theta)*w.
Thus if you rotate z = r*exp(i*phi) by its negative phase angle (-phi) , you arrive at z' = r*exp(i*phi)*exp(-i*phi) = r, the magnitude of z.
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Torsten
Torsten 2022년 8월 21일
If you diagonalize H with S,
D = S*H*S^(-1)
you get
f' = exp(S^(-1)*(i*t*D)*S)*f = (S^(-1)*exp(i*t*D)*S)*f
thus
S*f' = exp(i*t*D) * (S*f)
where D is a real diagonal matrix.
exp(i*t*D) is a diagonal matrix with exp(i*t*d(j)) on the diagonal.
Thus theta(j) is kind of t*d(j) with d(j) as j-th eigenvalue of H.
But I think Bruno can better comment on this from the application side.
Aisha Mohamed
Aisha Mohamed 2022년 8월 21일
Thanks very much Bruno Luong and Torsten. this explanation is very useful and helped me a lots.

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dpb
dpb 2022년 8월 20일
Basically, just what it says -- albeit somewhat wordily, perhaps... :)
A vector in 2D has X,Y components; a complex variable can be represented as a vector in a 2D plane with X-->Re, Y-->Im components.
In that plane, the magnitude is the vector from the origin to the point at which the intersection of the X (Re) and Y(Im) lines intersect; the angle of that vector represents the phase. By Pythagoras, the magnitude is abs() value, but if you compute only it, then you don't know what the two components were any more; you've gained the size but lost the phase (angle). Hence, all you can do then is plot a point on the X axis.

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