Conditions for this vectors to be linearly dependent Consider $m<n$ positive definite matrices $A_1\dots,A_m\in\mathbb{R}^{n\times n}$ which are linearly independent matrices. This is, that there are no $c_1,\dots,c_n\in\mathbb{R}$ such that $\sum_{i=1}^m c_iA_i = 0$ unless $c_i=0, \forall i=1,\dots,m$.
How can I find vectors $x\in\mathbb{R}^n$ different from $0$ such that the set of vectors
$$
A_1x,\dots,A_mx
$$
become linearly dependent? This is, that there exists  $d_1,\dots,d_n\in\mathbb{R}$ different from all $0$,such that $\sum_{i=1}^m d_iA_ix = 0$.
This question is motivated by an example as the following. Consider the matrices
$$
A_i = \begin{bmatrix}
M & 0_{3 \times 1}\\
0_{1\times 3} & a_i
\end{bmatrix}
$$
where $a_i\in\mathbb{R}$ and $M\in\mathbb{R}^{3\times 3}$, and $a_1,\dots,a_m$ are different from each other. Hence, if I take
$$
x = \begin{bmatrix}
x' \\
0
\end{bmatrix}
$$
with some $x'\in\mathbb{R}^3$, then $A_ix = Mx', \forall i=1,\dots, m$. Thus, in this case (which is something like a "trivial example") we have that the vectors $A_ix$ are linearly dependent.

We have that
$$d_1A_1x+d_2A_2x+\ldots+d_mA_mx=0 \iff (d_1A_1+d_2A_2+\ldots+d_mA_m)x=0$$
that is $x \in \ker(d_1A_1+d_2A_2+\ldots+d_mA_m)$ then the problem is equivalent to find $d_i$ such that $\operatorname{rank}(d_1A_1+d_2A_2+\ldots+d_mA_m)<n$.
