Complex conjugate of a 5 x 5 matrices?

In summary, to get the complex conjugate of a matrix, you simply conjugate each entry. To find the Hermitian conjugate or adjoint, you take the complex conjugate of each entry and then transpose the matrix. A 5*5 matrix example was provided to demonstrate this process. It is important to differentiate between the transpose and Hermitian conjugate, as they are not the same.
  • #1
ayalam
13
0
How do you do it?
 
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  • #2
To get the complex conjugate of any matrix you just conjugate each entry.
 
  • #3
Sorry I am doing something similar; converting a ket into a bra. wouldn't something change like order.
 
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  • #4
matt grime is completely right for "complex conjugate". However, you may be intending "Hermitian conjugate" or "adjoint". To find the Hermitian conjugate (adjoint) of a complex matrix you take the complex conjugate of each entry and take the transpose: rows become columns.
 
  • #5
Can i get a 5*5 matrices example
 
  • #6
[tex]A = \left(\begin{array}{ccccc}0 & i & 3 & 2 & 1 - 2i \\ -i & 7 & 6 & 4 & 0 \\
3 + i & 2 & 0 & -2i & 4 \\ 9 & 1 & i & 8 & 1 \\ 0 & 0 & 0 & 7i & 0\end{array}\right)[/tex] [tex]A^T = \left(\begin{array}{ccccc}0 & -i & 3 + i & 9 & 0 \\ i & 7 & 2 & 1 & 0 \\ 3 & 6 & 0 & i & 0 \\ 2 & 4 & -2i & 8 & 7i \\ 1 - 2i & 0 & 4 & 1 & 0\end{array}\right)[/tex] [tex]A^\dagger = \left(\begin{array}{ccccc}0 & i & 3 - i & 9 & 0 \\ -i & 7 & 2 & 1 & 0 \\ 3 & 6 & 0 & -i & 0 \\ 2 & 4 & 2i & 8 & -7i \\ 1 + 2i & 0 & 4 & 1 & 0\end{array}\right)[/tex]

I hope that's right and I've not made any mistakes.
 
  • #7
In Nylex's example, [tex]A^T[/tex] is the transpose and [tex]A^{\dagger}[/tex] is the Hermitian conjugate or adjoint.
Using the same example, the "complex conjugate" that was originally asked for, and matt grimes described, would be
[tex]A^* = \left(\begin{array}{ccccc}0 & -i & 3 & 2 & 1 + 2i \\ i & 7 & 6 & 4 & 0 \\3 - i & 2 & 0 & 2i & 4 \\ 9 & 1 & -i & 8 & 1 \\ 0 & 0 & 0 & -7i & 0\end{array}\right)[/tex]
 
  • #8
nice examples
 
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Related to Complex conjugate of a 5 x 5 matrices?

1. What is a complex conjugate of a 5 x 5 matrix?

The complex conjugate of a 5 x 5 matrix is a matrix with the same elements, but with the imaginary components of each element changed to their negative value.

2. How is the complex conjugate of a 5 x 5 matrix calculated?

The complex conjugate of a 5 x 5 matrix is calculated by taking the complex conjugate of each individual element in the matrix.

3. What is the purpose of finding the complex conjugate of a 5 x 5 matrix?

Finding the complex conjugate of a 5 x 5 matrix is useful in various mathematical operations, particularly in solving equations involving complex numbers.

4. Can a complex conjugate of a 5 x 5 matrix be a real matrix?

No, a complex conjugate of a 5 x 5 matrix will always contain complex numbers and cannot be a real matrix.

5. Is the complex conjugate of a 5 x 5 matrix unique?

Yes, the complex conjugate of a 5 x 5 matrix is unique and will always have the same elements in the same order as the original matrix.

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