To generalize lossless filters to the multi-input, multi-output (MIMO) case, we must generalize conjugation to MIMO transfer function matrices:
Theorem: A transfer function matrix is lossless if and only if its frequency-response matrix is unitary, i.e.,
Let denote the length output vector at time , and let denote the input -vector at time . Then in the frequency domain we have , which implies
We have thus shown that in the MIMO case, losslessness is equivalent to having a unitary frequency-response matrix. A MIMO allpass filter is therefore any filter with a unitary frequency-response matrix.
Note that is a matrix product of a times a matrix. If , then the rank must be deficient. Therefore, . (There must be at least as many outputs as there are inputs, but it's ok to have extra outputs.)