Is U Unitary if A' Preserves the Adjoint Relationship?

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  • Thread starter ognik
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In summary, we have shown that if a similarity transformation $U$ satisfies the adjoint relationship and is unitary, then it must be unitary itself.
  • #1
ognik
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A particular similarity transformation yields:
A' = UAU-1
(A')\(\displaystyle \dagger\) = UA\(\displaystyle \dagger\)U-1
if the adjoint relationship is preserved (A'\(\displaystyle \dagger\)=A\(\displaystyle \dagger\)') and det(U)=1, then Show that U must Be Unitary
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I think I've nearly got it as follows, but not sure my arguments are completely valid:
|det(U)|=1 is one of the conditions for U to be Unitary

From A' = UAU-1:
$ ({A}^{'})^{\dagger} = (UA{U}^{-1})^\dagger = (A{U}^{-1})^\dagger{U}^{\dagger} = ({U}^{-1})^\dagger {A}^{\dagger} {U}^{\dagger} $

So, from (A')\(\displaystyle \dagger\) = UA\(\displaystyle \dagger\)U-1 above:
$ ({U}^{-1})^\dagger{A}^{\dagger}{U}^{\dagger} = U{A}^{\dagger}{U}^{-1} $
Then equating the RH elements, $ {U}^{\dagger}={U}^{-1} $ which makes U unitary...provided we can also show the the LH elements are equal, IE show $ ({U}^{-1})^\dagger = U $
Now if U is Unitary, then $ ({U}^{-1})^\dagger = ({U}^\dagger)^\dagger = U $

This last step I don't think I have argued correctly, how would I improve the argument, or is there a better way of doing this proof? Thanks
 
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  • #2
for your help!Your reasoning is on the right track, but it can be simplified and made more rigorous. Here's a possible proof:

First, note that since $U$ is a similarity transformation, it must be invertible. Therefore, its adjoint $U^\dagger$ also exists.

Next, we have $(A')^\dagger = (UAU^{-1})^\dagger = (U^{-1})^\dagger A^\dagger U^\dagger$, where we have used the fact that the adjoint of a product is the product of the adjoints in the last step.

Now, since we are given that $(A')^\dagger = A^\dagger$, we can equate the two expressions to get $(U^{-1})^\dagger A^\dagger U^\dagger = A^\dagger$. This holds for any matrix $A$, so we can choose a specific matrix, say $A = I$, the identity matrix. This gives us $(U^{-1})^\dagger U^\dagger = I$.

Finally, we can take the adjoint of both sides to get $(U^\dagger)^\dagger (U^{-1})^\dagger = I^\dagger = I$. But we know that $U$ is unitary if and only if $U^\dagger U = I$, so we can rewrite this as $U^\dagger (U^{-1})^\dagger = I$. Comparing this to our previous equation, we see that $(U^{-1})^\dagger = U^\dagger$, and since $U^\dagger = U^{-1}$, this means that $(U^{-1})^\dagger = U$, as desired.

Note that this proof does not rely on the determinant condition, as the unitarity of $U$ is already built into the definition of a similarity transformation. So we do not need to explicitly show that $|det(U)| = 1$.

I hope this helps!
 

FAQ: Is U Unitary if A' Preserves the Adjoint Relationship?

What does it mean for a matrix to be unitary?

A unitary matrix is a square matrix whose complex conjugate transpose is equal to its inverse. In other words, the product of a unitary matrix and its conjugate transpose is equal to the identity matrix. This property ensures that the matrix is both orthogonal and norm preserving.

Why is it important to show that U must be unitary?

Unitary matrices have many important properties in mathematics and physics. They are used in quantum mechanics to represent transformations on quantum states, and in linear algebra to diagonalize other matrices. Showing that U must be unitary helps us understand the underlying structure and behavior of the matrix.

How can we prove that U must be unitary?

To prove that U must be unitary, we can use the definition of a unitary matrix and show that the product of U and its conjugate transpose is equal to the identity matrix. This can be done by multiplying the elements of the matrix and simplifying the resulting expression.

What are some applications of unitary matrices?

Unitary matrices are used in various fields, including quantum mechanics, signal processing, and data compression. They can also be used to solve systems of linear equations, diagonalize other matrices, and perform rotations and reflections in higher dimensions.

Are there any other properties of unitary matrices?

In addition to being orthogonal and norm preserving, unitary matrices also have the property that their eigenvalues have a magnitude of 1. This means that the matrix does not change the length of any vector it acts on. Unitary matrices also form a group under matrix multiplication, making them useful in group theory and abstract algebra.

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