Orthogonal Similarity and Pairs of Quadratic Forms原文及译文

原文：

Orthogonal similarity is a relation between two quadratic forms that preserves the angles between the subspaces defined by the forms. Two quadratic forms Q and Q' are said to be orthogonal similar if there exists an invertible matrix S such that S^TQS = Q'. In other words, the matrices representing the two forms are related by a congruence transformation.

Pairs of quadratic forms that are orthogonal similar have many important properties in common. For example, they have the same rank, determinant, and signature. They also have the same number of positive, negative, and zero eigenvalues. These properties make orthogonal similarity a powerful tool in the study of quadratic forms and their applications.

One important result related to orthogonal similarity is the Sylvester's law of inertia, which states that any two real symmetric matrices are orthogonal similar to diagonal matrices with the same number of positive, negative, and zero diagonal entries.

译文：

正交相似是两个二次型之间的一种关系，它保持了由这些二次型定义的子空间之间的角度关系。如果存在一个可逆矩阵S，使得S^TQS = Q'，则称二次型Q和Q'正交相似。换句话说，表示这两个二次型的矩阵通过一个合同变换相关联。

正交相似的二次型对具有许多重要的共同性质。例如，它们具有相同的秩、行列式和特征值符号。它们还具有相同数量的正、负和零特征值。这些性质使得正交相似成为研究二次型及其应用的强大工具。

与正交相似相关的一个重要结果是西尔维斯特的惯性定理，它指出任何两个实对称矩阵都可以通过正交相似变换转化为具有相同数量的正、负和零对角元素的对角矩阵。