Let $\mathcal{L}=({\mathcal{L}}_1,{\mathcal{L}}_2)$ be a list consisting of structural data for a matrix polynomial; here ${\mathcal{L}}_1$ is a sublist consisting of powers of irreducible (monic) scalar polynomials over the field $\mathbb{R}$, and ${\mathcal{L}}_2$ is a sublist of nonnegative integers. For an arbitrary such $\mathcal{L}$, we give easy-to-check necessary and sufficient conditions for $\mathcal{L}$ to be the list of elementary divisors and minimal indices of some real T-palindromic quadratic matrix polynomial. For a list $\mathcal{L}$ satisfying these conditions, we show how to explicitly build a real T-palindromic quadratic matrix polynomial having $\mathcal{L}$ as its structural data; that is, we provide a T-palindromic quadratic realization of $\mathcal{L}$ over $\mathbb{R}$. A significant feature of our construction differentiates it from related work in the literature; the realizations constructed here are direct sums of blocks with low bandwidth, that transparently display the spectral and singular structural data in the original list $\mathcal{L}$.

让$\mathcal{大号}=({\mathcal{大号}}_{\mathrm{1个}},{\mathcal{大号}}_{\mathrm{2个}})$是由矩阵多项式的结构数据组成的列表；这里${\mathcal{大号}}_{\mathrm{1个}}$是一个子列表，由场上的不可约（一元）标量多项式的幂组成$\mathbb{R}$， 和${\mathcal{大号}}_{\mathrm{2个}}$是非负整数的子列表。对于任意这样的$\mathcal{大号}$，我们给出易于检查的充分必要条件$\mathcal{大号}$是一些实T回文二次矩阵多项式的基本除数和最小索引的列表。对于列表$\mathcal{大号}$满足这些条件，我们展示了如何显式构建一个实数T回文二次矩阵多项式$\mathcal{大号}$作为其结构数据；也就是说，我们提供了一个T 回文二次实现$\mathcal{大号}$超过$\mathbb{R}$. 我们构建的一个显着特征将其与文献中的相关工作区分开来；这里构造的实现是低带宽块的直接总和，透明地显示原始列表中的光谱和奇异结构数据$\mathcal{大号}$.