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How do I span a vector space of $4\times 4$ matrices with real values by symmetric and skew symmetric matrices?

The basis of vector space of $4\times 4$ matrices has 16 elements, each containing one 1 and fifteen 0's. All I have to figure out is finding a combination of symmetric and skew symmetric matrices to get each of these elements.

Please just provide a hint.

Thank you in advance.

  • One approach you could take, rather than writing each element of the standard basis as a linear combination of symmetric and skew-symmetric matrices, is to find a basis of the subspace of symmetric matrices, and a basis of the subspace of skew-symmetric matrices, and then combine these two bases. – bradhd May 08 '13 at 02:36
  • Another approach you could take starts with the observation that if $A$ is a $4\times 4$ matrix, then $A+A^T$ is symmetric... – bradhd May 08 '13 at 02:37
  • Thanks @Brad for the insight. –  May 08 '13 at 02:43

2 Answers2

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Hint: For any square matrix $A$, one has $A = \frac{1}{2}(A+A^T) + \frac{1}{2}(A-A^T)$.

Alistair Savage
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1

Hint:

If $E_{ij}$ is the matrix whose $kl$-th entry is $\delta_{ik}\delta_{jl}$, consider the matrices of the form $E_{ij}\pm E_{ji}$.

Jared
  • 31,451