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Calculating angles between matrices

  1. Jul 13, 2011 #1
    Hey all,

    I was hoping someone could explain to me how to calculate the angle between matrices, ie. two square matrices

    [ 2 0
    0 -1]

    and

    [0 1
    1 3^(1/2)]

    under the inner product <A|B> = trace (A^TB)

    Also, how would you go about determining an angle between x and y when they are functions, ie. x = f(x) = x^2 +2 and y=(g(x)=x^3 -7x, under the inner product below:

    ⟨f |g⟩ =
    1
    ∫ f (x)g(x)dx.
    −1

    I already know how to determine angle using cos theta = (x^Ty)/ ||x|| ||y|| but does this only work for column and row matrices?
     
  2. jcsd
  3. Jul 13, 2011 #2
    Any time you have an inner product, you can use it to define angles between elements using the usual formula:
    [tex]
    \cos\theta(a,b) = \frac{\langle a | b \rangle}{\sqrt{\langle a | a \rangle \langle b | b \rangle}}
    [/tex]
    This works equally well for your matrices and for your functions.

    Whether that angle tells you anything useful for the problem you're trying to solve is another question. :)
     
  4. Jul 14, 2011 #3
    Thanks so much!
     
  5. Jul 14, 2011 #4

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    For "vectors in space", two or three dimensions, you can use trigonometry to prove that the dot product of two vectors, u and v, are given by [itex]u\cdot v= |u||v|cos(\theta)[/itex] where |u| and |v| are the lengths of the two vectors and [itex]\theta[/itex] is the angle between them. For more abstract vector spaces, we define the length of a vector, v, to be [itex]\sqrt{<v, v>}[/itex] and the angle between u and v to be given by
    [tex]cos(\theta)= \frac{<u, v>}{|u||v|}[/tex]
     
  6. Jul 17, 2011 #5
    In spicytaco's matrix example the angle is the same as the angle for the two vectors (2,0,0,-1) and (0,1,1, 3^1/2) in R^4 with usual euclidean norm.

    (2,0,0,-1).(0,1,1,3^1/2) = 0 + 0 + 0 - 3^1/2

    |(2,0,0,-1)| = 5^1/2
    |(0,1,1,3^1/2)| = 5^1/2

    so cos(theta) = (-3^1/2)/5 (theta ~= 110 degrees)

    this is why the inner product is defined as trace(A^T.B), for nxn matrices you get R^(n^2) euclidean space with two vectors defined by joining the columns of A and the other by the columns of B.

    For the functions example there isn't a "natural" analogy but the answer has been given above, you just plug in the formula for the inner product into the equation for cos(theta)
     
    Last edited: Jul 17, 2011
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