Scattering Matrix with Power Waves and Voltage Waves
The numbers in the matrix will be different. If you find the matrix with a particular normalization, you can compute it with a different normalization by multiplying. 

There are three normalizations commonly used. In books on circuit theory the matrix is often normalized to one ohm. 

For RF work with real devices, most scattering matrices are referred to 50 ohms. 

If you are working with cable tv or something, the normalization will be to 75 ohms, as the cables used with these devices have that characteristic impedance. 

The theorems involving the s matrix (for example, unitary=loosless) will be true no matter what the normalization used.