Most General Lossless Feedback Matrices

As shown in §H.13.3, an FDN feedback matrix is lossless if and only if its eigenvalues have modulus 1 and its eigenvectors are linearly independent.

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

where denotes the Hermitian conjugate (i.e., the complex-conjugate transpose) of . When is real (as opposed to complex), we may simply call it an orthogonal matrix, and we write , where 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.

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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.