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I Change of Basis and Unitary Transformations

  1. Feb 17, 2017 #1
    Say, we have two orthonormal basis sets ##\{v_i\}## and ##\{w_i\}## for a vector space A.

    Now, the first (old) basis, in terms of the second(new) basis, is given by, say,

    $$v_i=\Sigma_jS_{ij}w_j,~~~~\text{for all i.}$$

    How do I explicitly (in some basis) write the relation, ##Uv_i=w_i##, for some unitary matrix, ##U##?

    What is the relation between the matrix formed by the numbers ##S_{ij}## and ##U##?
     
  2. jcsd
  3. Feb 17, 2017 #2

    mathman

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    The [tex]S_{ij} [/tex] form a matrix. U is its inverse.
     
  4. Feb 17, 2017 #3
    How do I see this explicitly?
     
  5. Feb 17, 2017 #4

    Stephen Tashi

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    Can you see how the simultaneous equations

    can be expressed as an equation involving matrices?

    It's the same idea as expressing

    1) ##ax + by = c##
    2) ##dx + ey = f##

    as

    ##\begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} c \\f \end{pmatrix}##
     
  6. Feb 17, 2017 #5
    Yes, but where I got confused was that the entries of the column vectors themselves are vectors.
    $$v_j=\sum_jS_{ij}w_i,~~~~\text{for all}~ j.$$
    The ##v_j~'s ## and ##w_i~'s## in the equation above are not numbers but vectors.

    But, I think, I have figured it out.

    Let, U be a linear transformation on the vector space A, and ##\{v_j\}## be a basis. Then,
    $$Uv_j=\sum_iU_{ij}v_i$$
    Now, we define
    $$\sum_iU_{ij}v_i=w_j\\ \implies v_j=U^{-1}w_j$$

    Now, if we write the old basis vectors ##\{v_j\}## in terms of the new basis,
    $$v_j=\sum_iS_{ij}w_i=Sw_j$$
    Comparing the two equations for ##v_j##, we see that ##S=U^{-1}.##

    In fact, we can write the the operators, ##U## and ##S## as outer products of the basis vectors,
    $$U=\sum_j v_j\otimes v_j\\ S=\sum_i w_i \otimes w_i.$$

    Do you think this makes sense? Or are there still some lacunae in my understanding?
     
  7. Feb 18, 2017 #6

    Stephen Tashi

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    You need to have a "##v_i##" instead of a "##v_j##" on the left hand side of that equation.

    Yes, the terms involved in that sum , ##S_{i,j} w_i##, are vectors, but we aren't yet dealing with any representation of ##w_i## as a vector of numbers. So, effectively, ##w_i## plays the role of a variable, just like the typical uses of "##x##" and "##y##".

    Yes, those equations ( considering all ##i##) define a linear transformation ##U## expressed in the ##v##-basis.

    To interpret each equation as involving a matrix multiplication ##U v_j = b_j##, we must adopt some convention about how ##v_j## is represented as a column vector. Using "##v_j##" as ambiguous notation, we could say that ##v_j## represents a column vector of "variables" that are all zeroes except for the variable "##v_j##" in its ##j##th entry. Or we could say that ##v_j## ( in ##v##-basis) will be a column vector with a 1 in the ##j##th entry of the column and zeroes elsewhere. So interpreting the left hand side of the equation as matrix multiplication does involve a some convention about how ##v_j## is represented as a column vector.

    We must also adopt a similar convention about the column vector on the right hand side. We might say it represents a linear combination of vectors from the ##v##-basis and that the ##k##th entry of the column vector on the right side the equation represents the coefficient of ##v_k## in the linear combination.


    It isn't clear what you are defining. Are you defining ##U## or are you defining the ##w_j## ?

    If we take the ##w##-basis as given, you are defining ##U## as a matrix whose ##j##th column consists of the coefficients needed to express ##w_j## as a linear combination of vectors in the ##v##-basis.

    However, the usual way to express the ##w##-basis in terms of the ##v##-basis in matrix form would be to consider the column vector ##w = (w_1,w_2,w_3,..)## as the ##w##-basis expressed as a vector of "variables", and the column vector ##w = (v_1,v_2,v_3,...)## as the ##v##-basis expressed a vector of variables, and to define the matrix ##U## to satisfy ##w = Uv##.

    This definition would imply ##w_i = \sum_j U_{i,j} v_j ##, so the summation is over the column index "##j##" instead of the row index "##i##".
     
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