LWE with secret matrix (Reverse LWE?) I was wondering if there is a version of LWE with secret matrices and public seed vectors? Would it be as hard as the popular definition of LWE?
The LWE assumption tells you that $(\mathbf{a},\mathbf{a}\mathbf{s}+e)$ looks random for a hidden random vector $\mathbf{s}$, a random vector $\mathbf{a}$ and small error $e$.

Invoking the LWE assumption k times, you get that $(\mathbf{a},\mathbf{a}\mathbf{s}_1+e_1,\mathbf{a}\mathbf{s}_2+e_2,\ldots,\mathbf{a}\mathbf{s}_k+e_k)$ is indistinguishable from $(\mathbf{a},b_1,b_2,\ldots,b_k)$ where the $b_i$'s are random values.

So $(\mathbf{a},\mathbf{S}\mathbf{a}+\mathbf{e})$ is indistinguishable from $(\mathbf{a},\mathbf{b})$, and if an adversary has advantage $\varepsilon$ in distinguishing those 2 distributions, there is an adversary that has advantage $\frac{1}{k}\varepsilon$ in distinguishing LWE samples from uniform, where $k$ is the dimension of the vector $\mathbf{e}$.