Given a set of $p$ symmetric (real) matrices, the Orthogonal Joint Diagonalization (OJD) problem consists of finding an orthonormal basis in which the representation of each of these $p$ matrices is as close as possible to a diagonal matrix. We argue that when the matrices are of large dimension, then the natural generalization of this problem is to seek an orthonormal basis of a certain subspace that is a near eigenspace for all the matrices in the set. We refer to this as the problem of ``partial joint diagonalization of matrices.'' The approach proposed first finds this approximate common near eigenspace and then proceeds to a joint diagonalization of the restrictions of the input matrices in this subspace. A few solution methods for this problem are proposed and illustrations of its potential applications are provided.

翻译：给定一组 $p$ 个对称（实）矩阵，正交联合对角化问题旨在寻找一组标准正交基，使得这些 $p$ 个矩阵在该基下的表示尽可能接近对角矩阵。我们认为，当矩阵维度较大时，该问题的自然推广是寻找某个子空间的标准正交基，使其成为集合中所有矩阵的近似公共特征子空间。我们将此问题称为“矩阵的部分联合对角化”。所提出的方法首先找到这个近似的公共特征子空间，然后在该子空间内对输入矩阵的限制进行联合对角化。本文提出了几种解决该问题的方法，并举例说明了其潜在应用。