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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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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2014-03-23 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University