Verify matrix is unitary

  • #1
8
0
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.
 

Attachments

  • Unitary Matrix verification.JPG
    Unitary Matrix verification.JPG
    14.6 KB · Views: 2,298

Answers and Replies

  • #2
38
3
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}##.
 
  • Like
Likes andphy
  • #3
8
0
That helps thank you.
 
  • #4
Fredrik
Staff Emeritus
Science Advisor
Gold Member
10,851
413
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
HallsofIvy
Science Advisor
Homework Helper
41,833
963
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
8
0
Right the product will result in the inverse:

1 0
0 1

correct ?
 
Last edited:
  • #7
Fredrik
Staff Emeritus
Science Advisor
Gold Member
10,851
413
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
8
0
sorry meant to say identity matrix (not inverse) - thank you.
 

Related Threads on Verify matrix is unitary

  • Last Post
Replies
2
Views
741
Replies
2
Views
830
  • Last Post
Replies
2
Views
3K
  • Last Post
Replies
1
Views
2K
  • Last Post
Replies
1
Views
9K
  • Last Post
Replies
2
Views
671
Replies
16
Views
1K
S
Replies
4
Views
617
Replies
30
Views
1K
  • Last Post
Replies
6
Views
2K
Top