Suppose I have a matrix $Xin mathbb{R}^{ntimes n}$, such that

- $X$ is symmetric
- Do not know the rank

I want design a **matrix function** $f(X,Q)in mathbb{R}^{ntimes n}$ with $Q = qq^T$ and $qin mathbb{R}^n$, such that

- $f(X,Q)$ is symmetric
- $text{tr} (f(X,Q))=0$
- rank$(f(X,Q))$ is $2$.

For

- we can design $QX+XQ$
- I only know the property $Omega Q + QOmega^T$ with $Omega$ a skew-symmetric matrix (since tr$(QOmega)=q^TOmega q=0$.
- Choose $f(X,Q)=QX+XQ$. The basis for its column space is ${q, Xq}$

However, I have no idea how to combine all of them. Can anyone help me this?

Sincerely appreciate this help.