Stability of the FDN is assured when some norm  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
where denotes the spectral norm of .
It can be shown  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
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.
Allpass from Two Combs
Single-Input, Single-Output (SISO) FDN