Does the adjoint of an outer product equal to itself?

In summary, an outer product is a mathematical operation that involves multiplying two vectors to create a matrix. The resulting matrix has elements that are the products of each element in the first vector with each element in the second vector. An adjoint, also known as the conjugate transpose, is a mathematical operation on a matrix that involves taking the transpose of the matrix and then taking the complex conjugate of each element. Yes, the adjoint of an outer product always equals itself because the operation does not change the structure of the matrix. There are no exceptions to this rule, and it has important implications in quantum mechanics and linear algebra. It allows for simplification of calculations and proofs, and helps to establish relationships between different mathematical operations.
  • #1
onelastdance
6
0
So (|a><a|)† = |a><a|
 
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  • #2
Why don't you try to prove it??
 
  • #3
A good start might be to try a few test cases. Write down a small (2- or 3-dimensional) complex vector ##|a \rangle## and calculate its outer product with itself. Then check each matrix element to see if the real parts of ##| a \rangle \langle a|## are symmetric and the imaginary parts are anti-symmetric.
 

1. What is an outer product?

An outer product is a mathematical operation that involves multiplying two vectors to create a matrix. The resulting matrix has elements that are the products of each element in the first vector with each element in the second vector.

2. What is an adjoint?

An adjoint, also known as the conjugate transpose, is a mathematical operation on a matrix that involves taking the transpose of the matrix and then taking the complex conjugate of each element.

3. Does the adjoint of an outer product always equal itself?

Yes, the adjoint of an outer product always equals itself. This is because the adjoint operation does not change the structure of the matrix, it simply changes the values of the elements. Since the outer product matrix is symmetric, taking the transpose and conjugate does not change it.

4. Are there any exceptions to the adjoint of an outer product equaling itself?

No, there are no exceptions to this rule. The adjoint of an outer product will always equal itself, regardless of the size or values of the vectors used to create the outer product.

5. What is the significance of the adjoint of an outer product equaling itself?

The fact that the adjoint of an outer product equals itself has important implications in quantum mechanics and linear algebra. It allows for simplification of calculations and proofs, and helps to establish relationships between different mathematical operations.

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