(mathbb{R}^{ntimes m}), (mathbb{C}^{ntimes m}) and (mathbb {H}^{ntimes m}) denote (ntimes m) matrices with entries from (mathbb{R}), (mathbb{C}) and (mathbb{H}), respectively.
Let (mathbb{M}_{n}(mathcal{M})) denote the linear space of (ntimes n) matrices with entries (x_{ij}inmathcal{M}), (i,j=1,2,ldots,n).
In the rest of the paper, the set of real numbers, the vector of n-tuples of real numbers and the space of (ntimes n) matrices with real entries are denoted by ℜ, (Re^{n}), and (Re^{ntimes n}), respectively.
Throughout the paper, (Re^{n}) and (Re^{ntimes n}) represent the vectors of n-tuples of real numbers and the space of (ntimes n) matrices with real entries, respectively, and (mathbb{N}) is the set of nonnegative integers.
As a special case of Example 3.1, one can consider the following matrix equation, which can also be found in [19]: X-A_{1}^{ast}XA_{1}-cdots-A_{m}^{ast}XA_{m}=Q, where Q is a positive definite matrix and (A_{1},ldots, A_{m}) are arbitrary (ntimes n) matrices with (sum_{n=1}^{m}|A_{n}|<1) with (pgeq1).
We consider the singular value decomposition (SVD) of a matrix A of rank r A=USV^{H},quad S=operatorname{diag}bigl({sigma_{i}}bigr), 1leq ileq r, where U and V are (mtimes r) and (ntimes r) matrices with orthonormal columns, respectively, and (sigma_{i}) is the positive singular values.
The dual basis (f_j) is defined by begin{aligned} langle f_j,e_ell rangle = delta _{jell } quad text { i.e. } sum _{r=1}^{N} f_{jr} e_{ell r} = delta _{j ell } end{aligned} (11.27 If we think of E, F as the (Ntimes N) matrices with (F_{jr}=(f_j)_r,, E_{jr} = (e_j)_r), then (11.27) says that (FE^T={varvec{1}}).
We consider linear systems of delay differential equations with periodic coefficients of the form frac{d}{dt}y(t) = mu A t y(t) + B t y(t - tau),quad t > 0, (1) where (A t)), (B t)) are ((ntimes n)) matrices with T-periodic continuous entries; i.e., A(t+T) equiv A t), qquad B(t+T) equiv B t), (tau> 0) is the time delay, and (mu> 0) is a parameter.
We obtain the spectral norms of some (ntimes n) C matrices, with the aid of Theorem 3 (see Table 1).
