### 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

*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

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