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?
 

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

What is the Conjugate Transpose of a Complex Matrix?

The conjugate transpose of a complex matrix, also known as the Hermitian transpose, is obtained by taking the transpose of the matrix and then taking the complex conjugate of each entry. If \( A \) is a complex matrix, its conjugate transpose is denoted as \( A^* \) or \( A^\dagger \).

How Do We Prove That the Conjugate Transpose of a Conjugate Transpose Returns the Original Matrix?

To prove that \((A^*)^* = A\), consider a complex matrix \( A \). The conjugate transpose \( A^* \) is formed by transposing \( A \) and then taking the complex conjugate of each element. Applying the conjugate transpose operation again to \( A^* \) means transposing \( A^* \) and taking the complex conjugate of each element, which returns the original matrix \( A \).

What is the Relationship Between the Conjugate Transpose of a Product of Two Matrices and the Conjugate Transposes of the Individual Matrices?

The relationship is given by the property \((AB)^* = B^*A^*\). This can be proved by considering the definition of the conjugate transpose. For matrices \( A \) and \( B \), the element \((AB)_{ij}\) is the dot product of the \( i \)-th row of \( A \) and the \( j \)-th column of \( B \). Taking the conjugate transpose of \( AB \) involves transposing and then conjugating, which results in the product of \( B^* \) and \( A^* \).

How Can We Prove That the Conjugate Transpose of a Sum of Matrices is the Sum of Their Conjugate Transposes?

To prove that \((A + B)^* = A^* + B^*\), consider matrices \( A \) and \( B \). The element \((A + B)_{ij}\) is \( A_{ij} + B_{ij} \). Taking the conjugate transpose of \( A + B \) involves transposing and then conjugating each element, which results in \( A^* + B^* \), since the operations of addition and taking the conjugate transpose commute.

What is the Conjugate Transpose Property of a Scalar Multiple of a Matrix?

The property is \((\alpha A)^* = \overline{\alpha} A^*\), where \( \alpha \) is a complex scalar and \( \overline{\alpha} \) is its complex conjugate. This can be proved by noting

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