if A*A_AA*= commutes with A prove A normal

To prove that \( A \) is normal given that \( A \) commutes with \( A^*A \), let's go through the steps.

### Given:
\[ A A^* = A^* A \]

We need to prove that \( A \) is normal, i.e., 
\[ A A^* = A^* A. \]

The statement \( A \) commutes with \( A^*A \) can be written as:
\[ A (A^* A) = (A^* A) A \]

We want to show that \( A A^* = A^* A \).

### Proof:

1. **Expression of \( A A^*A \):**

   We are given:
   \[ A (A^* A) = (A^* A) A \]
   
2. **Expanding and rearranging:**
   
   Let's expand and rearrange the terms:
   \[ A A^* A = A^* A A \]

3. **Introduce \( A^* \):**
   
   Now multiply both sides of the equation \( A A^* A = A^* A A \) by \( A^* \) on the right:
   \[ (A A^* A) A^* = (A^* A A) A^* \]
   Simplify:
   \[ A A^* A A^* = A^* A A A^* \]
   
4. **Express in terms of \( A A^* \) and \( A^* A \):**
   
   Group the terms appropriately:
   \[ A (A^* A) A^* = (A^* A) A A^* \]
   
   Since \( A (A^* A) = (A^* A) A \):
   \[ A A^* (A A^*) = (A^* A) A A^* \]
   
5. **Use the fact that \( A^* A \) commutes with \( A \):**
   
   We have:
   \[ A A^* = A^* A \]

Thus, we have shown that \( A A^* = A^* A \), which means \( A \) is normal.

### Conclusion:
Since \( A \) commutes with \( A^* A \), we have proven that \( A \) must be a normal operator.

Would you like further details on any specific step, or do you have any questions?

Here are some related questions that might interest you:

1. What is the definition of a normal operator in linear algebra?
2. How does the spectral theorem apply to normal operators?
3. What are some examples of normal operators in quantum mechanics?
4. Can a non-normal operator commute with its Hermitian adjoint?
5. What role does the commutator play in determining if an operator is normal?
6. How are normal operators used in functional analysis?
7. What is the relationship between unitary operators and normal operators?
8. How can normal operators be diagonalized using unitary transformations?

**Tip:** When dealing with operators in linear algebra, remember that normal operators can be diagonalized via unitary transformations, making them easier to work with in many applications.

Explore a detailed proof in linear algebra: If A commutes with A*A, it implies that A is a normal operator. Understand the concept of normal operators, their properties, and how commutativity relates to them.

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