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

• kokolo
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.
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,

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,
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/

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}##

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?

DaveE
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}##

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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