# Products of PSD matrices that equal an orthogonal matrix

You have k symmetric real n-by-n matrices A_1...A_k, each bounded by some parameter eps < 1/2 in spectral norm.

You know that the product (Id+A_1)(Id+A_2)...(Id+A_k) equals an orthogonal matrix Y.

What can you say about the norm of A_1+...+A_k? (any norm - frobenius, spectral, something else). note that if n=1 (the scalar case) then clearly it's bounded by O(k eps^2) (by passing from (1+A_i) to exp(A_i+O(A_i^2)), then solving a trivial equation).

what can you say about the distance of Y from being the identity matrix (e.g. how low can the trace of Y be?)