Show How to Write A as B + iC: Hermitian Operators

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An arbitrary operator A can be expressed as A = B + iC, where B and C are Hermitian operators. This is demonstrated by rewriting A using the formula A = (A + A†)/2 + (A - A†)/2. The first term, (A + A†)/2, is Hermitian, while the second term, (A - A†)/2, is anti-Hermitian, leading to the conclusion that A can be decomposed into Hermitian components. The discussion emphasizes that this decomposition holds true for any operator with an involution J, allowing for a similar representation. Understanding this relationship is fundamental in the study of Hermitian operators.
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How do I show that an arbitrary operator A can be writte as A = B + iC where B and C are hermitian?
 
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Rewrite A as follows:

A = \frac{(A+A^{\dagger})}{2} + \frac{(A-A^{\dagger})}{2}

Do you see why you can write A like that?
And can you carry on?
 
Last edited:
anytime anywhere you have an involution J you can alwaYS WRiTE ANYTHIng AS

x = (x+JX)/2 + (X-JX)/2,

where X+JX is invariant under J, and X-JX is anti-invariant under J.

this is what lies beneath this fact.
 

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