The truncated moment problem asks to characterize finite sequences of real numbers that are the moments of a positive Borel measure on ${\mathbb{R}}^n$. Its tracial analog is obtained by integrating traces of symmetric matrices and is the main topic of this article. The solution of the bivariate quartic tracial moment problem with a nonsingular $7\times 7$ moment matrix ${\mathcal{M}}_2$ whose columns are indexed by words of degree 2 was established by Burgdorf and Klep, while in our previous work we completely solved all cases with ${\mathcal{M}}_2$ of rank at most 5, split ${\mathcal{M}}_2$ of rank 6 into four possible cases according to the column relation satisfied and solved two of them. Our first main result in this article is the solution for ${\mathcal{M}}_2$ satisfying the third possible column relation, i.e., ${\mathbb{Y}}^2=\mathbb{1}+{\mathbb{X}}^2$. Namely, the existence of a representing measure is equivalent to the feasibility problem of certain linear matrix inequalities. The second main result is a thorough analysis of the atoms in the measure for ${\mathcal{M}}_2$ satisfying ${\mathbb{Y}}^2=\mathbb{1}$, the most demanding column relation. We prove that size 3 atoms are not needed in the representing measure, a fact proved to be true in all other cases. The third main result extends the solution for ${\mathcal{M}}_2$ of rank 5 to general ${\mathcal{M}}_n$, $n\ge 2$, with two quadratic column relations. The main technique is the reduction of the problem to the classical univariate truncated moment problem, an approach which applies also in the classical truncated moment problem. Finally, our last main result, which demonstrates this approach, is a simplification of the proof for the solution of the degenerate truncated hyperbolic moment problem first obtained by Curto and Fialkow.

截断矩问题要求刻画实数的有限序列，这些实数是正Borel测度的矩。 ${\mathbb{[R}}^{ñ}$。通过集成对称矩阵的痕迹可以得到其类似的痕迹，这是本文的主题。具有非奇异值的二元四次分叉弯矩问题的求解$7\times 7$ 矩矩阵 ${\mathcal{中号}}_2$ Burgdorf和Klep建立了以2级单词索引的列，而在我们之前的工作中，我们完全解决了所有情况 ${\mathcal{中号}}_2$ 最多排名5，分裂 ${\mathcal{中号}}_2$根据满足的列关系，将等级6分为四种可能的情况，并解决了其中两种情况。本文的第一个主要结果是解决方案${\mathcal{中号}}_2$ 满足第三个可能的列关系，即 ${\mathbb{ÿ}}^2=\mathbb{1个}+{\mathbb{X}}^2$。即，代表度量的存在等同于某些线性矩阵不等式的可行性问题。第二个主要结果是对测量${\mathcal{中号}}_2$ 满意的 ${\mathbb{ÿ}}^2=\mathbb{1个}$，最苛刻的列关系。我们证明在表示量度中不需要大小为3的原子，事实在所有其他情况下都成立。第三个主要结果扩展了解决方案${\mathcal{中号}}_2$ 从第5级到一般 ${\mathcal{中号}}_{ñ}$， $ñ\ge 2$，具有两个二次列关系。主要技术是将问题简化为经典单变量截断矩问题，该方法也适用于经典截断矩问题。最后，我们最后的主要结果证明了这种方法，是简化了由Curto和Fialkow首先获得的简并截断双曲矩问题的证明。