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

[kokolo]

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

 

[fresh_42]

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/

 

[kokolo]

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/

$Y^* X= \overline{X^* Y}$

 

[fresh_42]

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?

 

[kokolo]

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 $\overline{X^* \cdot Y}$

 

[fresh_42]

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?