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Math Help - basis for Snn

  1. #1
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    basis for Snn

    Another one. The M n,n is a vector space, and that Snn( the set of n x n symmetric matrices) is a subspace
    Find a basis for the space of all 2 x 2 symmetric matrices.
    B)Find a basis for the space of all 3 x 3 symmetric matrices
    C) Determine the dimensions of the space of n x n symmetric ( Hint: this essentially a counting argument.From examining the bases in the first two parts determine what goes into the basis required.
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  2. #2
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    Re: basis for Snn

    Mnn is a space of all m x n matrices Symmetric matrix Snn are in M and is a subset of all symmetric matrices, exchanges rows and columns such as A=AT (A,B) are in Symmetric matrices such as (aij=aji or bij=bji) an example of this is A+B==aij+bij, therefore (A+B)T = (aji+bji)=(BT)+(AT) hope this helps.
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  3. #3
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    Re: basis for Snn

    a 2x2 symmetric matrix is of the form:

    M = \begin{bmatrix}a&b\\b&c \end{bmatrix}

    this suggests the basis set:

    B = \left\{\begin{bmatrix}1&0\\0&0 \end{bmatrix}, \begin{bmatrix}0&1\\1&0 \end{bmatrix}, \begin{bmatrix}0&0\\0&1 \end{bmatrix} \right\}

    linear independence and spanning should be clear (that is: you should be able to prove these on demand).

    for 3x3 symmetric matrices, prove the following is a basis set:

    B' = \left\{\begin{bmatrix}1&0&0\\0&0&0\\0&0&0 \end{bmatrix}, \begin{bmatrix}0&1&0\\1&0&0\\0&0&0 \end{bmatrix}, \begin{bmatrix}0&0&1\\0&0&0\\1&0&0 \end{bmatrix}, \begin{bmatrix}0&0&0\\0&1&0\\0&0&0 \end{bmatrix}, \begin{bmatrix}0&0&0\\0&0&1\\0&1&0 \end{bmatrix}, \begin{bmatrix}0&0&0\\0&0&0\\0&0&1 \end{bmatrix}\right\}

    does this suggest what to do for the nxn case? (hint: consider the diagonal and off-diagonal entries separately).
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  4. #4
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    Re: basis for Snn

    Thank can anyone help with part C given Deveno's answer.
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  5. #5
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    Re: basis for Snn

    another hint: you should have n(n+1)/2 matrices. where did i get this number?
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