I read awhile back that the set of continuous real valued functions from $\mathbb{R} \to \mathbb{R} $ has a direct sum decomposition into subspaces of strictly even and odd functions. Any such function $f$ could then be uniquely written in terms of even and odd parts by the formulas $\text{even}(x)=\dfrac{f(x)+f(-x)}{2}$ and $\text{odd}(x)=\dfrac{f(x)-f(-x)}{2}$ where $\text{even}(x)+\text{odd}(x)=f(x)$.

Recently I stumbled into a similar claim for $n\times n$ matrices, but instead of even and odd, the set of square matrices has a direct sum representation in terms of symmetric and skew symmetric parts. So given any square matrix $A$ we can write $B=\dfrac{A+A^T}{2}$ and $C=\dfrac{A-A^T}{2}$ as the symmetric and skew symmetric representations where $A=B+C$.

Does this mean that the notion of symmetric and skew symmetric have a connection to the notion of even and odd? Do they represent a generalization of even/odd, or is there another idea that generalizes these two ideas?

  • $\begingroup$ You are very observant and this is something I had not thought of before! The connection is what happens when you change the sign of $x$ in $f(x)$. If it is even, there is no sign change and if it is odd the sign changes. Now in the case of matrices, what happens if you interchange $x$ and $y$ in $x^T M y$? If M is symmetric, then no change. If it is skew symmetric, the sign changes. Nice insight. I am going to use it! $\endgroup$ – user44197 Dec 31 '13 at 21:32

Let $V$ be any real vector space, and let $T$ be any operator on $V$ with $T$ not the identity but $T^2$ the identity. Given any $v$ in $V$, we have $v=u+w$ with $T(u)=u$ and $T(w)=-w$, namely, $u=(v+T(v))/2$ and $w=(v-T(v))/2$.

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    $\begingroup$ I guess that you could add that the vector space now can be the space of real functions or the space of matrices. $\endgroup$ – Phira Jan 1 '14 at 9:00

You can think of the representation of a complex matrix as the sum of a Hermitian and skew-Hermitian matrix as the analogy to the real and imaginary part of a complex number. It is even more apparent when considering their spectral properties.


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