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Symmetric matrices help please!

  1. Nov 10, 2009 #1
    symmetric matrices.... help please!

    hi can someone tell me...how to correctly use the 10 axioms..
    for example: does the set of all 3x3 symmetric matrices form a subspace of a 3x3 matrices under the normal operations of matrix addition and multiplication?
    I don't really get how to prove this..
     
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  3. Nov 10, 2009 #2

    HallsofIvy

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    Re: symmetric matrices.... help please!

    The only things you need to prove are that the set is closed under addition and scalar multiplication. Suppose
    [tex]A= \begin{bmatrix}a & b & c \\b & d & e \\c & e & f\end{bmatrix}[/tex]
    and
    [tex]B= \begin{bmatrix} u & v & w \\ v & x & y \\w & y & z\end{bmatrix}[/tex]
    two symmetric matrices. Is A+ B symmetric? Is [itex]\alpha A[/itex] symmetric for any number [itex]\alpha[/itex]?
     
  4. Nov 13, 2009 #3
    Re: symmetric matrices.... help please!

    I thought a subspace had to also contain the zero vector (not empty), or am I remembering that wrong?
     
  5. Nov 14, 2009 #4

    HallsofIvy

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    Re: symmetric matrices.... help please!

    Well, yes. Thanks for the correction. I should have included that- but the 0 matrix is rather trivially symmetric isn't it?

    My point is that you do NOT have to prove all "10 axioms". Given a subset of a vector space, things such as "u+ v= v+ u" follow from the fact that they are true of the entire vector space and so true for any subset.

    To prove that "U is a subspace of vector space V" you need to prove:
    1) U is closed under vector addition.
    2) U is closed under scalar multiplication.
    and the one I forgot:
    3) U is non-empty.
     
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