Is the Transpose Conjugate of a Unitary Matrix Equal to the Identity Matrix?

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In summary, the conversation is discussing the properties of a unitary matrix and how to verify if a given matrix is unitary. The speaker explains that a unitary matrix should have its transpose conjugate equal to its inverse, and that this can be checked by multiplying the matrix with its conjugate. The conversation then moves on to discussing specific matrices and their inverses, with the final conclusion being that the product of a matrix and its conjugate should result in the identity matrix.
  • #1
andphy
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Hi,

A unitary matrix should have it transpose conjugate equal to its inverse. Please confirm that this statement is correct and check attached matrix as they are not equal and in doubt if I did correctly.

Thanks.
 

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  • #2
Hi,
suppose to have an unitary matrix (##U \in \mathbb{C}^{n \times n}##, so that ##U^{\dagger}=U^{-1}##), if you want to verify the unitariety of your matrix, just check if ##UU^{\dagger}= \mathbb{I}##.
 
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  • #3
That helps thank you.
 
  • #4
Are you saying (in the attached file) that ##\begin{pmatrix}i & 0\\ 0 & 1\end{pmatrix}## is the inverse of ##\begin{pmatrix}1 & 0\\ 0 & i\end{pmatrix}##? It's not. To see this, just multiply these two matrices together.
 
  • #5
You have at one point, that the "transpose conjugate" of [itex]\begin{bmatrix}1 & 0 \\ 0 & i\end{bmatrix}[/itex] is [itex]\begin{bmatrix}1 & 0 \\ 0 & -i\end{bmatrix}[/itex]. That is correct and that is the inverse matrix.

Below that, you have "inverse matrix" and [itex]\begin{bmatrix} i & 0 \\ 0 & 1\end{bmatrix}[/itex]. I don't know where that came from!
 
  • #6
Right the product will result in the inverse:

1 0
0 1

correct ?
 
Last edited:
  • #7
andphy said:
Right the product will result in the inverse:

1 0
0 1

correct ?
What product? The product of the two matrices in post #5 is ##\begin{pmatrix}i & 0 \\ 0 & i\end{pmatrix}##. As HallsofIvy said, the inverse of ##\begin{pmatrix}1 & 0\\ 0 & i\end{pmatrix}## is ##\begin{pmatrix}1 & 0\\ 0 & -i\end{pmatrix}##. The product of these two matrices is the identity matrix.
 
  • #8
sorry meant to say identity matrix (not inverse) - thank you.
 

What is a unitary matrix?

A unitary matrix is a square matrix whose conjugate transpose is equal to its inverse. In other words, it is a matrix that, when multiplied by its conjugate transpose, results in the identity matrix.

How do you verify if a matrix is unitary?

To verify if a matrix is unitary, you can follow these steps:
1. Calculate the conjugate transpose of the matrix
2. Multiply the original matrix by its conjugate transpose
3. If the result is the identity matrix, then the matrix is unitary.

Can a non-square matrix be unitary?

No, a unitary matrix must be square. This is because only square matrices have a conjugate transpose and can be multiplied by themselves to result in the identity matrix.

Are all orthogonal matrices unitary?

Yes, all orthogonal matrices are unitary. This is because the conjugate transpose of an orthogonal matrix is equal to its inverse, making it satisfy the definition of a unitary matrix.

What is the significance of unitary matrices in science?

Unitary matrices are important in many areas of science, including quantum mechanics, signal processing, and linear algebra. They are used to represent and manipulate complex numbers, and have applications in quantum computing and cryptography. In quantum mechanics, unitary matrices describe the evolution of quantum states over time and are essential in understanding quantum systems.

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