The problem of approximate joint diagonalization of a collection of matrices arises in a number of diverse engineering and signal processing problems. This problem is usually cast as an optimization problem, and it is the main goal of this publication to provide a theoretical study of the corresponding cost-functional. As our main result, we prove that this functional tends to infinity in the vicinity of rank-deficient matrices with probability one, thereby proving that the optimization problem is well posed. Secondly, we provide unified expressions for its higher-order derivatives in multilinear form, and explicit expressions for the gradient and the Hessian of the functional in standard form, thereby opening for new improved numerical schemes for the solution of the joint diagonalization problem. A special section is devoted to the important case of self-adjoint matrices.

翻译：矩阵集合的近似联合对角化问题出现在众多工程与信号处理领域。该问题通常被表述为一个优化问题，本文的主要目标是对相应的代价函数进行理论研究。作为核心结论，我们证明了该函数在秩亏矩阵附近以概率一趋于无穷，从而证实了优化问题的适定性。其次，我们以多重线性形式给出了该函数高阶导数的统一表达式，并以标准形式给出了函数梯度与Hessian矩阵的显式表达式，这为求解联合对角化问题的新型改进数值方案开辟了道路。本文专设章节讨论了自伴矩阵这一重要情形。