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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:

$\displaystyle 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).

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