Hermitian Conjugate of Matrix Explained

In summary, the Hermitian Conjugate of a matrix is equal to its transpose conjugate, and this can be seen through the property (AB)^{+} = B^{+} A^{+}. This is not commutative, as shown by the different results for (AB)^{T} and (A^T B^T).
  • #1
raintrek
75
0
Simple question, and pretty sure I already know the answer - I just wanted confirmation,

Considering the Hermitian Conjugate of a matrix, I understand that

[tex]A^{+} = A[/tex] where [tex]A^{+} = (A^{T})^{*}[/tex]

Explicitly,

[tex](A_{nm})^{*} = A_{mn}[/tex]

Would this mean that for a matrix of A, where A is

a b
c d

that

a b
c d

=

a* c*
b* d*

=

A11 A12
A21 A22

=

A11* A21*
A12* A22*

Thanks for the clarification!
 
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  • #2
And can I also ask why this seems to be a general property of the Hermitian Conjugate?

[tex](AB)^{+} = B^{+} A^{+}[/tex]

rather than

[tex](AB)^{+} = A^{+} B^{+}[/tex]
 
  • #3
for your first post, you have done correct.

a b
c d

becomes

a* c*
b* d*

when you do hermitian conjugate of it.

And
[tex](AB)^{\dagger} = B^{\dagger} A^{\dagger}[/tex]

Follows from
[tex](AB)^{T} = B^{T} A^{T}[/tex]

Very easy to prove
 
  • #4
As for
[tex](AB)^{\dagger} = B^{\dagger} A^{\dagger}[/tex]
and
[tex](AB)^{T} = B^{T} A^{T}[/tex]

remember that multiplication of matrices is NOT commutative.
With [itex](AB)^{T} = B^{T} A^{T}[/itex] we have [itex](AB)^T(AB)= (A^T)(B^T B)(A)= A^T A= I[/itex]. If we tried, instead, [itex](A^TB^T)(AB)[/itex] we would have [itex](A^T)(B^T A)(B)[/itex] and we can't do anything with that.
 

1. What is the Hermitian conjugate of a matrix?

The Hermitian conjugate of a matrix is the complex conjugate of its transpose. It is denoted as AH or A* and is obtained by taking the complex conjugate of each element in the matrix and then taking the transpose of the resulting matrix.

2. What is the significance of the Hermitian conjugate in matrix operations?

The Hermitian conjugate plays a crucial role in matrix operations involving complex numbers. It allows us to define inner products, orthogonality, and unitarity of complex vectors and matrices, which are essential concepts in quantum mechanics and other fields. It also enables us to define the adjoint of a linear operator, which is used in solving systems of linear equations.

3. How is the Hermitian conjugate related to the complex conjugate?

The Hermitian conjugate and the complex conjugate are closely related, as the Hermitian conjugate is essentially the complex conjugate of a matrix. However, the Hermitian conjugate also involves taking the transpose of the complex conjugate, which makes it a more general operation. In other words, the complex conjugate only involves replacing each element with its complex conjugate, while the Hermitian conjugate also involves changing the order of the elements.

4. What are the properties of the Hermitian conjugate?

Some of the key properties of the Hermitian conjugate are:

  • (AH)H = A (The Hermitian conjugate of the Hermitian conjugate of a matrix is the matrix itself).
  • (AB)H = BHAH (The Hermitian conjugate of a product of matrices is the product of the Hermitian conjugates in the reverse order).
  • (λA)H = λ*AH (The Hermitian conjugate of a scalar multiple of a matrix is equal to the complex conjugate of the scalar multiplied by the Hermitian conjugate of the matrix).

5. How is the Hermitian conjugate used in quantum mechanics?

In quantum mechanics, the Hermitian conjugate is used to define the adjoint of a linear operator, which is crucial in solving the Schrödinger equation and other quantum mechanical equations. It is also used to define the inner product of complex vectors and matrices, which is necessary for calculating probabilities and other quantities in quantum mechanics.

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