How Can We Prove the Conjugate Transpose Property of Complex Matrices?

In summary, the conversation discusses proving the relationship between complex conjugate matrices and transposed matrices. The formula to be proved is (Y^*) * X = complex conjugate of {(X^*) * Y}, where ^* represents complex conjugation. The conversation also mentions the use of the transpose of a matrix and the adjoint matrix in the proof. Finally, the summary provides a helpful formula for solving the proof.
  • #1
kokolo
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TL;DR Summary: For every Complex matrix proove that: (Y^*) * X = complex conjugate of {(X^*) * Y}

Here (Y^*) and (X^*) is equal to complex conjugate of (Y^T) and complex conjugate of (X^T) where T presents transponse of matrix
I think we need to use (A*B)^T= (B^T) * (A^T) and
Can you help me proove this cause I'm really stuck,
Thanks in advance
 
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  • #2
kokolo said:
TL;DR Summary: For every Complex matrix proove that: (Y^*) * X = complex conjugate of {(X^*) * Y}

Here (Y^*) and (X^*) is equal to complex conjugate of (Y^T) and complex conjugate of (X^T) where T presents transponse of matrix
I think we need to use (A*B)^T= (B^T) * (A^T) and
Can you help me proove this cause I'm really stuck,
Thanks in advance
What do you know? What does ^* mean? Can you prove it for a single complex number, a ##1\times 1## matrix?

By the way: Here is explained how you can type formulas on PF: https://www.physicsforums.com/help/latexhelp/
 
  • #3
fresh_42 said:
What do you know? What does ^* mean? Can you prove it for a single complex number, a ##1\times 1## matrix?

By the way: Here is explained how you can type formulas on PF: https://www.physicsforums.com/help/latexhelp/
## Y^* X= \overline{X^* Y}##
 
  • #4
I have difficulties understanding what this is all about. Say ##\overline{X}## means the complex conjugate, ##X^T## means the transposed matrix, and ##X^\dagger=\overline{X}^T## the adjoint matrix (conjugate and transposed). Also, please write the multiplication ##X\cdot Y## with a dot. With these notations, what do you need to prove?
 
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  • #5
fresh_42 said:
I have difficulties understanding what this is all about. Say ##\overline{X}## means the complex conjugate, ##X^T## means the transposed matrix, and ##X^\dagger=\overline{X}^T## the adjoint matrix (conjugate and transposed). Also, please write the multiplication ##X\cdot Y## with a dot. With these notations, what do you need to prove?
##Y^* \cdot X=\overline{X^* \cdot Y}## where ##Y^*=\overline{Y^T}## and ##X^*=\overline{X^T}## and
complex conjugate matrix is ##\overline{X^* \cdot Y}##
 
  • #6
You have ##(X \cdot Y)^T=Y^T\cdot X^T## and ##(X\cdot Y)^*=\overline{X\cdot Y}^T=(\overline{X}\cdot\overline{Y})^T=\overline{Y}^T\cdot \overline{X}^T=Y^*\cdot X^*.##

Does this help?
 

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