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!