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Prove that any square matrix can be written as the sum of a symmteric and a skew-symm

  1. Nov 9, 2009 #1
    1. The problem statement, all variables and given/known data
    Prove that any square matrix can be written as the sum of a symmetric and a skew-symmetric matrix


    2. Relevant equations

    For symmetric [tex]A=A^{T}[/tex]
    For scew-symmetric [tex]A=-A^{T}[/tex]

    3. The attempt at a solution
    Not sure where to begin. Using algebra didn't work. Got powers and nothing cancelled.

    p.s is there an inbuilt template for matrix in latex?

    Thanks
    Tom
     
  2. jcsd
  3. Nov 9, 2009 #2

    Hurkyl

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    Re: Prove that any square matrix can be written as the sum of a symmteric and a skew-

    Have you tried some explicit examples?

    Could you explain what you wanted to try with algebra? And demonstrate what you actually tried?
     
  4. Nov 9, 2009 #3
    Re: Prove that any square matrix can be written as the sum of a symmteric and a skew-

    well i set up the matrix
    A =
    a b
    c d

    and
    [tex]A^{T} = \frac{a c}{b d}[/tex]


    and i multiplied the two together, but that doesn't get you anywhere?

    I've been reading through a book, and the last question presented in that topic. I have the answer on the following page

    A= 0.5(A+A^T) + 0.5(A-A^T)

    but I'm not sure how they come to that answer. I mean I can see that the transforms cancel each other out, but how did they get there?

    Thanks :)
     
  5. Nov 10, 2009 #4

    HallsofIvy

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    Re: Prove that any square matrix can be written as the sum of a symmteric and a skew-

    If f(x) is any function at all, then g(x)= f(x)+ f(-x) is an "even" function, because g(-x)= (-x)= f(-x)= f(-(-x))= f(x)+ f(-x)= fe(x), and h(x)= f(x)- f(-x) is an odd function, because h(-x)= f(-x)-f(-(-x))= -(f(x)- f(-x)). Adding those two functions, g(x)+ h(x)= 2f(x) so g(x)/2= (f(x)+ f(-x))/2 and h(x)/2= (f(x)-f(-x))/2 are even and odd functions that sum to f- they are the "even" and "odd" parts of f. Do you see the similarity with your problem?
     
  6. Nov 10, 2009 #5
    Re: Prove that any square matrix can be written as the sum of a symmteric and a skew-

    How many free parameters are there in a M times M matrix? How many are there if the matrix is symmetric or antisymmetric?
     
  7. Nov 10, 2009 #6

    D H

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    Re: Prove that any square matrix can be written as the sum of a symmteric and a skew-

    Denote B and C to represent the symmetric and skew-symmetric parts of the given matrix A: A=B+C

    Taking the transpose of A, AT=BT+CT. Now using the fact that B is symmetric and C is skew-symmetric, AT=B-C. The problem here is to solve for B and C. Writing the equations for A and AT as

    [tex]\aligned
    B+C&=A \\
    B-C&=A^T
    \endaligned[/tex]

    So, two linear equations in two unknowns, and a particularly easy one to solve at that.

    [tex]\aligned
    B&=(A+A^T)/2 \\
    C&=(A-A^T)/2
    \endaligned[/tex]
     
  8. Nov 11, 2009 #7
    Re: Prove that any square matrix can be written as the sum of a symmteric and a skew-

    Ahh I think I see DH!
    If we add B and C, we get A!
    [tex]A^{T} = B^{T}+C^{T}[/tex]

    As we have square matrices we can say [tex]B^{T} = B[/tex] and [tex]C^{T} = -C[/tex]
    this means
    [tex]A^{T} = B-C[/tex]
    solving that with
    A = B + C

    gives us your bottom 2 equations (the symmetric and skew-symmetric matrices), which we add up to give a matrice.


    Hallofivy, I can see what your kinda getting at!

    clamtrox I don't see what you mean by parameters. A matrix holds elements, are these the parameters? Can you expand?

    Thanks
    Tom
     
  9. Nov 12, 2009 #8
    Re: Prove that any square matrix can be written as the sum of a symmteric and a skew-

    Yes yes, I was just thinking about a more high-flying explanation. It obviously works like DH explained.

    But yeah, a physicist looks immediately how many degrees of freedom there are. Since a M times M matrix contains M^2 independent functions (independent elements, parameters, degrees of freedom, DOF's), then one would expect that a symmetric M times M and an antisymmetric M times M matrix would have the same number of DOF's combined.
     
  10. Nov 16, 2009 #9
    Re: Prove that any square matrix can be written as the sum of a symmteric and a skew-

    I was just looking back over this and I was wondering if you could expand on it

    I know that an even function is that which f(x) = f(-x) - like a quadratic
    An odd function is f(-x) = -f(x) - like x³

    When hitting this line, my brain turns to mush:
    g(-x)= (-x)= f(-x)= f(-(-x))= f(x)+ f(-x)= fe(x)

    so a negative value x is inputted into function g. G is defined as f(x) + f(-x). If f(x) is odd then f(x) + f(-x) = 0, but I'm not sure what
    g(-x)= (-x)= f(-x)= f(-(-x))= f(x)+ f(-x)= fe(x)
    is telling me. I don't even know what the e does. Can you say a function in number e.

    Never really taught such maths in that kind of "comprehensive" manner. I'm a sucker for pretty pictures ;)

    Thanks
    Tom
     
    Last edited: Nov 16, 2009
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