# reference request – rank of a linear combination of matrices

Let $$A_1,…, A_s in M_n(mathbb{R})$$ be symmetric matrices and suppose they are linearly independent over $$mathbb{R}$$. This means that
$$m = min_{(c_1, …, c_s) in mathbb{R}^s backslash {0}} rank( sum_{i=1}^s c_i A_i ) > 0$$
I am interested in the question how large can $$m$$ be?
I am not sure where to start looking… if someone could point me to a good reference it’s very appreciated! Also any comments are appreciated!