Exploiting the explicit bijection between the density of singular values and the density of eigenvalues for bi-unitarily invariant complex random matrix ensembles of finite matrix size we aim at finding the induced probability measure on $j$ eigenvalues and $k$ singular values that we coin $j,k$-point correlation measure. We fully derive all $j,k$-point correlation measures in the simplest cases for matrices of size $n=1$ and $n=2$. For $n>2$, we find a general formula for the $1,1$-point correlation measure. This formula reduces drastically when assuming the singular values are drawn from a polynomial ensemble, yielding an explicit formula in terms of the kernel corresponding to the singular value statistics. These expressions simplify even further when the singular values are drawn from a P\'{o}lya ensemble and extend known results between the eigenvalue and singular value statistics of the corresponding bi-unitarily invariant ensemble.

翻译：利用有限矩阵尺寸下双酉不变复随机矩阵系综中奇异值密度与特征值密度之间的显式双射，我们旨在寻找由$j$个特征值和$k$个奇异值诱导的概率测度，并将其定义为$j,k$点相关测度。针对尺寸为$n=1$和$n=2$的矩阵，我们完整推导了所有$j,k$点相关测度。当$n>2$时，我们给出了$1,1$点相关测度的一般公式。该公式在假设奇异值取自多项式系综时显著简化，并基于与奇异值统计量对应的核函数得到显式表达式。当奇异值取自Pólya系综时，这些表达式进一步简化，并推广了相应双酉不变系综中特征值与奇异值统计量之间的已知结果。