Unitary matrix and preservation of vector norm in arbitrary basis

In summary, the conversation discusses the topic of unitary matrices and their preservation of length in a particular basis. A proof is provided using the fact that A†A = I, but this proof is only valid for vectors defined in an orthonormal basis. The use of this proof in the book is questioned and a potential counterexample is presented. The conversation concludes with the clarification that unitary matrices only preserve the usual inner product, and not all possible inner products.
  • #1
D_Tr
45
4
Hi PF people!
I am not sure my question can elegantly fit in the template, but I 'll try.

Homework Statement



I am self-studying the 8th chapter of "Mathematical Methods for Physics and Engineering", 3rd edition by Riley, Hobson, Bence. In the section about unitary matrices, it is stated that:

"A unitary matrix represents, in a particular basis, a linear operator that leaves the norms (lengths) of complex vectors unchanged. If y = Ax is represented in some coordinate system by the matrix equation y = Ax then <y|y> is given in this coordinate system by: y†y = x†A†Ax = x†x = <x|x>.

The above proof used the fact that A†A = I since A is unitary. BUT: Isn't the definition of the inner product used in this proof only valid for vectors defined in an orthonormal basis? In the same chapter, the inner product of two vectors in an arbitrary basis is straightforwardly proved to be:
<a|b> = a†Gb, where G is a square matrix and Gij = <ei|ej>, where ei, ej are basis vectors. When the components are with respect to an orthonormal basis we can calculate the inner product by just multiplying the corresponding components and adding the products (G becomes the identity matrix), but in an non orthonormal basis this is not the case.
Question: Do unitary matrices preserve length in a non-orthonormal basis? If yes, how do we prove it? Why does the book use the formula for the orthonormal basis? Some sources I found on google just do what the book does whitout reference to the type of basis used.

Homework Equations



The relevant equations are the ones in 1, I do not think that anything else is needed.

The Attempt at a Solution



In a non orthonormal basis, we have:

<y|y> = y†Gy = (Ax)†G(Ax) = x†A†GAx

<x|x> is x†Gx in an arbitrary basis. Additionally, G is hermitian, since <ei|ej> = <ej|ei>*, and A is unitary.

Can we prove that x†A†GAx = x†Gx or does this simply not hold? I am really baffled... What I was thinking was the following example: If we construct a matrix that transforms vector [a, b, c], into [b, c, a], this matrix is unitary (the matrix actually is [0 1 0, 0 0 1, 1 0 0]), since times its hermitian conjugate (just its transpose, since it is real) gives the identity matrix. The permuted vector definitely has the same norm in an orthonormal basis, but in an orthogonal basis where we chose one basis vector to be much longer than the others, wouldn't this permutation generally change the vector's length? So if what I am thinking is correct, this unitary matrix changes the vector's norm in a non orthonormal basis...

Thanks for reading :)
 
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  • #2
You are correct, a unitary matrix only preserves the usual inner product. So if we define the inner product like

[tex]<X,Y> = X^T Y[/tex]

then it follows for a unitary matrix that ##<Ax,Ay> = <x,y>##.

If you change your inner product to something of the forms

[tex]<X,Y> = X^T G Y[/tex]

then unitary matrices will not preserve the inner product anymore. So the question in your thread title only works for the usual inner product.
 
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  • #3
Thank you for your time! I was going crazy because I could not, for the life of me, figure out what could I possibly be missing... In the book it was stated "...in some coordinate system...", so it seems that this was an oversight on the writers' part, because only one kind of inner product is discussed in the chapter.
 

Related to Unitary matrix and preservation of vector norm in arbitrary basis

1. What is a unitary matrix?

A unitary matrix is a square matrix whose conjugate transpose is equal to its inverse. In other words, a unitary matrix is a matrix that preserves the length of vectors and angles between them.

2. How does a unitary matrix preserve vector norm?

A unitary matrix preserves vector norm by multiplying a vector by the unitary matrix does not change its length. This is because the unitary matrix is orthogonal, meaning it does not alter the magnitude of the vector.

3. What is the significance of preserving vector norm in arbitrary basis?

Preserving vector norm in arbitrary basis is important because it allows us to use any basis we want to represent a vector without changing its length. This is useful in many applications, such as quantum mechanics, where different basis are often used to represent the same vector.

4. How is the preservation of vector norm related to the eigenvalues of a unitary matrix?

The eigenvalues of a unitary matrix have a magnitude of 1, which means they do not alter the length of a vector when multiplied by the unitary matrix. This is why a unitary matrix preserves vector norm.

5. Can a non-unitary matrix preserve vector norm?

No, a non-unitary matrix cannot preserve vector norm. A non-unitary matrix may change the length of a vector when multiplied by it, whereas a unitary matrix maintains the vector's length.

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