Is $a b^T$ + $b a^T$ positive semidefinite if $a^T b > 0$

Question

Let $a$ and $b$ are $m \times 1$ constant vectors such that $a^T b > 0$. I am wondering if the matrix $a b^T$+$b a^T$ is positive semidefinite. Thanks

Answer 1

No. You can make up plenty of examples. You want vectors $a,b,x$ so that, for example, $a\cdot b > 0$, $a\cdot x>0$ and $b\cdot x<0$. Then $(ab^\top + ba^\top)x\cdot x = 2(a\cdot x)(b\cdot x)$ will be negative. For example: $$a=\begin{bmatrix} 1\\1 \end{bmatrix}, \quad b=\begin{bmatrix} -2\\3 \end{bmatrix}, \quad x=\begin{bmatrix} 2\\1 \end{bmatrix}$$ should yield $(ab^\top+ba^\top)x\cdot x = -42$.