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PHYSICAL INTERPRETATION OF HERMITIAN CONJUGATE

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kashyapramaa

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I WANTED TO KNOW THE PHYSICAL INTERPRETATION OF HERMITIAN CONJUGATE
WHICH IS EXPLAINED ONLY FROM A MATHEMATICAL VIEWPOINT EVERYWHERE.

THANKS IN ADVANCE
 

rrumpf

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I am not sure I can completely answer your question, but this is my best "guess." A hermitian matrix is usually hermitian because it was derived from a description of a real physical system. Compare this to the Fourier transform of a real signal. The negative and positive frequencies come in complex conjugate pairs. Similarly, if you want to calculate energy in a signal, you multiple its complex amplitude by the conjugate of its complex amplitude (i.e. P(A)=AA*). For this reason, you will sometimes see the hermitian conjugate in calculations related to energy. You may also see it in what are called similarity transforms. In other cases, this operation can be interchanged with the standard transpose. For example, in the computation of left and right eigensystems, sometimes you will see the standard transpose used while other times you will see the hermitian used.

I will add this, if you are using MATLAB, the (') operate is a hermitian (conjugate transpose) operation. A standard transpose is performed as (.'). That is

B = A'; %Hermitian operation
C = A.'; %standard transpose

I have no idea why Mathworks did it this way, but I it is a very common pitfall.

-Tip
 
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