Does Matrix projection impacts centralization

Question

I have a dataset that each sample is a matrix $A_{m\times n}$, and I have N samples in my dataset. I want to centralize my data across my samples, meaning that for each element in the matrix, I want to subtract the mean of that element across all samples from it. If I do that and then use a projection matrix P and project all the matrices, are the projected matrices still centralized? Based on my research I think if the projection matrix is orthogonal then yes they are still centralized, if that's true how can I prove it?