Can I simplify the following vector? Let $\boldsymbol{a}$ and   $\boldsymbol{b}$ be $n \times 1$ vectors and $M_1, M_2, ..., M_n$ be $n\times n$ matrices.
Is there a way to write the following vector:
\begin{equation}
v = \begin{bmatrix}
a^T M_1 b
\\
a^T M_2 b
\\
\vdots
\\
a^T M_n b
\end{bmatrix}
\end{equation}
in such a way that reads $v = a^T M_{new} b$, where $M_{new}$ is just a manipulation of the $M_i$ matrices?
I know I can do:
\begin{equation}
v = \begin{bmatrix}
a^T M_1 
\\
a^T M_2 
\\
\vdots
\\
a^T M_n 
\end{bmatrix}
b
\end{equation}
But then I cannot get the vector $a$ "out of the brackets".

You could write
$$
v = \pmatrix{a^T \\& \ddots & \\&&a^T} \pmatrix{M_1\\ \vdots \\ M_n}b.
$$
If you like, you could write the matrix on the left compactly using the fact that
$$
\pmatrix{a^T \\& \ddots & \\&&a^T} = a^T \otimes I,
$$
where $I$ denotes a size $n$ identity matrix and $\otimes$ denotes the Kronecker product.
