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

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SUMMARY

An arbitrary operator A can be expressed as A = B + iC, where both B and C are Hermitian operators. This is achieved by rewriting A using the formula A = (A + A†)/2 + (A - A†)/2. The discussion emphasizes that for any involution J, any operator x can similarly be decomposed into components that are invariant and anti-invariant under J. This foundational principle underpins the representation of operators in quantum mechanics.

PREREQUISITES
  • Understanding of Hermitian operators in quantum mechanics
  • Familiarity with operator notation and adjoint operators (A†)
  • Knowledge of involution concepts in linear algebra
  • Basic principles of quantum mechanics and operator theory
NEXT STEPS
  • Study the properties of Hermitian operators in quantum mechanics
  • Learn about the implications of operator adjoints and their significance
  • Explore the concept of involution in linear algebra
  • Investigate the role of operators in quantum state representation
USEFUL FOR

Quantum physicists, mathematicians specializing in linear algebra, and students studying operator theory in quantum mechanics will benefit from this discussion.

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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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