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Unitary matrix and preservation of vector norm in arbitrary basis

  1. Apr 6, 2014 #1
    Hi PF people!
    I am not sure my question can elegantly fit in the template, but I 'll try.

    1. The problem statement, all variables and given/known data

    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.

    2. Relevant equations

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

    3. 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 :)
     
    Last edited: Apr 6, 2014
  2. jcsd
  3. Apr 6, 2014 #2

    micromass

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    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.
     
  4. Apr 6, 2014 #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.
     
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