We address the problem of finding the closest matrix \(\tilde{\varvec{U}}\) to a given \(\varvec{U}\) under the constraint that a prescribed second-moment matrix \(\tilde{\varvec{P}}\) must be matched, i.e. \(\tilde{\varvec{U}}^{\mathrm {T}}\tilde{\varvec{U}}=\tilde{\varvec{P}}\). We obtain a closed-form formula for the unique global optimizer \(\tilde{\varvec{U}}\) for the full-rank case, that is related to \(\varvec{U}\) by an SPD (symmetric positive definite) linear transform. This result is generalized to rank-deficient cases as well as to infinite dimensions. We highlight the geometric intuition behind the theory and study the problem’s rich connections to minimum congruence transform, generalized polar decomposition, optimal transport, and rank-deficient data assimilation. In the special case of \(\tilde{\varvec{P}}=\varvec{I}\), minimum-correction second-moment matching reduces to the well-studied optimal orthonormalization problem. We investigate the general strategies for numerically computing the optimizer and analyze existing polar decomposition and matrix square root algorithms. We modify and stabilize two Newton iterations previously deemed unstable for computing the matrix square root, such that they can now be used to efficiently compute both the orthogonal polar factor and the SPD square root. We then verify the higher performance of the various new algorithms using benchmark cases with randomly generated matrices. Lastly, we complete two applications for the stochastic Lorenz-96 dynamical system in a chaotic regime. In reduced subspace tracking using dynamically orthogonal equations, we maintain the numerical orthonormality and continuity of time-varying base vectors. In ensemble square root filtering for data assimilation, the prior samples are transformed into posterior ones by matching the covariance given by the Kalman update while also minimizing the corrections to the prior samples.

我们解决了在给定的第二矩矩阵\（\ tilde {\ ）的约束下找到最接近给定\（\ varvec {U} \）的矩阵\（\ tilde {\ varvec {U}} \）的问题。 varvec {P}} \）必须匹配，即 \（\ tilde {\ varvec {U}} ^ {\ mathrm {T}} \ tilde {\ varvec {U}} = \ tilde {\ varvec {P}} \）。我们为全秩情况获得了唯一全局优化器\（\ tilde {\ varvec {U}} \）的闭式公式，该公式与\（\ varvec {U} \）相关通过SPD（对称正定）线性变换。该结果被推广到秩不足的情况以及无穷大的情况。我们强调了该理论背后的几何直觉，并研究了问题与最小同余变换，广义极分解，最优输运和秩不足数据同化的丰富联系。在\（\ tilde {\ varvec {P}} = \ varvec {I} \）的特殊情况下，最小校正第二矩匹配简化为经过充分研究的最佳正交归一化问题。我们研究了数值计算优化器的一般策略，并分析了现有的极坐标分解和矩阵平方根算法。我们修改并稳定了先前被认为不稳定的两个牛顿迭代，用于计算矩阵平方根，以便现在可以将其用于有效地计算正交极性因子和SPD平方根。然后，我们使用具有随机生成矩阵的基准案例来验证各种新算法的更高性能。最后，我们完成了混沌状态下的随机Lorenz-96动力系统的两个应用程序。在使用动态正交方程的简化子空间跟踪中，我们保持时变基向量的数值正交性和连续性。在用于数据同化的集合平方根滤波中，通过匹配卡尔曼更新给出的协方差，同时将对先前样本的校正最小化，将先前样本转换为后验样本。