Most General Lossless Feedback Matrices

As shown in §C.15.3, an FDN feedback matrix $\mathbf{A}_N$ is lossless if and only if its eigenvalues have modulus 1 and its $N$ eigenvectors are linearly independent.

A unitary matrix $Q$ is any (complex) matrix that is inverted by its own (conjugate) transpose:

$$Q^{-1} = Q^H,$$

where $Q^H$ denotes the Hermitian conjugate (i.e., the complex-conjugate transpose) of $Q$ . When $Q$ is real (as opposed to complex), we may simply call it an orthogonal matrix, and we write $Q^{-1} = Q^T$ , where $T$ denotes matrix transposition.

All unitary (and orthogonal) matrices have unit-modulus eigenvalues and linearly independent eigenvectors. As a result, when used as a feedback matrix in an FDN, the resulting FDN will be lossless (until the delay-line damping filters are inserted, as discussed in §3.7.4 below).