Table 10 Notation

${\mathbb{R}}^n$

n-dimensional vector space of real-valued vectors

${\mathbb{C}}^n$

n-dimensional vector space of complex-valued vectors

$|x|:={max}_{1\le i\le n}|x_i|$

maximum norm for a vector $x\in {\mathbb{C}}^n$

C[0,1]

space of continuous real vector-valued functions on [0,1]

$C^p[0,1]$

space of p-times continuously differentiable real vector-valued functions on [0,1]

$\parallel y\parallel :={max}_{t\in [0,1]}|y(t)|$

maximum norm for a function y∈C[0,1]

${\parallel y\parallel }_{\delta }:={max}_{t\in [0,\delta ]}|y(t)|$

norm restricted to the interval [0,δ], δ>0

$|A|:={max}_{1\le i\le m}{\sum }_{j=1}^n|a_{ij}|$

induced operator norm for a matrix $A\in {\mathbb{C}}^{m\times n}$

$X_{+}$

invariant subspace associated with the eigenvalues with positive real parts

$X_0^{(e)}$

space spanned by eigenvectors associated with eigenvalues λ = 0

$X_{-}$

invariant subspace associated with the eigenvalues with negative real parts

$X_0^{(h)}$

space spanned by principal eigenvectors associated with the eigenvalue λ = 0

S

orthogonal projection onto $X_{+}$

R

orthogonal projection onto $X_0^{(e)}$

P: = R + S

projection onto $X_{+}\oplus X_0^{(e)}$

Q: = I − P

projection onto $X_{-}\oplus X_0^{(h)}$

Z: = R + H

orthogonal projection onto $X_0^{(e)}\oplus X_0^{(h)}$

N

orthogonal projection onto $X_{-}$

H

orthogonal projection onto $X_0^{(h)}$