for all , where
denotes the identity
matrix, and
denotes the Hermitian transpose
(complex-conjugate transpose) of
:

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

or

Integrating both sides of this equation with respect to
yields that the total energy in equals the total energy out, as
required by the definition of losslessness.

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.)