### FDN Stability

Stability of the FDN is assured when some *norm* [451] of
the state vector
decreases over time when the input signal is
zero [220, ``Lyapunov stability theory'']. That is, a
sufficient condition for FDN stability is

for all , where denotes the norm of , and

for all , where denotes the

*norm*, defined by

The *matrix norm* corresponding to any vector norm
may be defined for the matrix
as

*spectral norm*. Thus, Eq.(2.13) can be restated as

where denotes the spectral norm of .

It can be shown [167] that the spectral norm of a matrix
is given by the largest singular value of
(``
''), and that this is equal to the
square-root of the largest eigenvalue of
, where
denotes the matrix transpose of the real matrix
.^{3.11}

Since every *orthogonal matrix*
has spectral norm
1,^{3.12} a wide variety of stable
feedback matrices can be parametrized as

An alternative stability proof may be based on showing that an FDN is a special case of a passive digital waveguide network (derived in §C.15). This analysis reveals that the FDN is lossless if and only if the feedback matrix has unit-modulus eigenvalues and linearly independent eigenvectors.

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Single-Input, Single-Output (SISO) FDN