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I know how to show that $W_1$ is a subspace but I don't know what it wants me to think by saying "assume $F$ is not of characteristic $2$". I know that when $F$ is not of characteristic $2$, it means either $x+x+...+x=0$ (the number of $x$'s is greater than $2$) or characteristic $0$. I don't see its use in the problem. How can I show that $M_{n\times n}(F)$ is a direct sum of $W_1$ and $W_2$?

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shinobi20
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  • This is just because if it is of char2 the dimension would be 3 instead of 1 – Quality Apr 30 '15 at 05:40
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    In characteristic $2$ a matrix is skew-symmetric if and only if it is symmetric. –  Apr 30 '15 at 05:40
  • If an arbitrary field has a characteristic 2, does it mean that it only contains 1 and 0 and satisfy the properties of a characteristic 2 field? I don't get it, if an arbitrary field has characteristic 0, does it mean 1+1 is 2 and not 0? – shinobi20 Apr 30 '15 at 05:47
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    No, the field can contain other elements. The characteristic just describes the smallest field contained in the field: the field generated by 1. The field can be bigger than its characteristic subfield. Characteristic 0 means the smallest field is the rationals, whereas characteristic 2 implies the smallest field is of size 2 (and therefore $1+1=2=0$). – jgon Apr 30 '15 at 06:15
  • see https://math.solverer.com/library/stephen_friedberg/linear_algebra/exercise_1-3-28?utm_source=stackexchange.com&utm_medium=link – togrultopal Dec 08 '21 at 11:19

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In characteristic $2$, the skew-symmetric and symmetric matrices are the same, since $M^T=-M=M$. So for the decomposition given to make sense, which need a characteristic that isn't $2$.

To show $M_{n\times n}=W_1\oplus W_2$, fix some $S\in M_{n\times n}$ and let $$A=\frac{S+S^T}{2},$$ $$B=\frac{S-S^T}{2}.$$ Verify that $A$ is symmetric, $B$ is skew-symmetric, and $A+B=S$. Also verify that $W_1\cap W_1 = 0$.

Potato
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  • Thanks for that, I think I got it, but the only thing that still confuses me is how can M=-M in characteristic 2? And how can someone think of constructing a matrix such that A and B is what you wrote? I mean, I know that what you wrote works, but how did you think of writing it that way? – shinobi20 Apr 30 '15 at 07:15
  • @shinobi20 1. Characteristic $2$ means that $2A=0$ for any element $A$. Subtracting $A$ from both sides gives $A=-A$. – Potato Apr 30 '15 at 07:18
  • I said to myself, suppose we have a decomposition $S=A+B$, where $A$ is symmetric and $B$ is skew-symmetric. Then $S^T=A-B$, and you can solve for $A$ and $B$.
    • – Potato Apr 30 '15 at 07:19
  • Thank you so much for the explanation, I really understood the problem now and the concept of characteristic. The book doesn't have a section discussing about characteristics that is why I'm so puzzled. By the way, is this a kind of proof that you find a way to construct an object that satisfies the problem but then you just state in your formal proof that "Let this be this and that be that"? That is working backwards? – shinobi20 Apr 30 '15 at 07:39
  • @shinobi20 Yes, I just worked backwards. But as you note, you should work forwards in your proofs. – Potato Apr 30 '15 at 07:40
  • I see, thank you very much for the help. Appreciated it! – shinobi20 Apr 30 '15 at 07:41
  • @Potato : It is being alleged at meta that you are not able to receive any notification of my comments at the URL below. Is that true? http://math.stackexchange.com/questions/1258577/error-function-etymology-why-the-name – Michael Hardy Apr 30 '15 at 22:33
  • @MichaelHardy I did not receive any notification of your comments, and I do not see any of your comments there now. Is there a problem? – Potato Apr 30 '15 at 23:51
  • I see that the question has been reopened and all of the comments including mine have been deleted. I have no idea how the latter was done. – Michael Hardy Apr 30 '15 at 23:54