Simultaneously sparsifying symmetrical real matrices

Question

Suppose that we have $m\in N_+$ symmetrical real matrix $A_i,i=1,2...m,A_i\in R^{n\times n}$. Can we find an orthogonal matrix $T\in R^{n\times n}, TT'=T'T=I$ such that all of $B_i=T'A_iT$ are sparse matrices? In this problem setting, $m\ll n$ and sparsity means the number of non-zeros elements in a $n\times n$ matrix are $o(n^2)$.