正交矩阵的若干性质研究.pdf

摘要：正交矩阵是实数域上一类十分特殊的矩阵，它作为矩阵的一个重要分支，无论是在自然科学领域还是在社会科学领域中都有非常广泛的应用，尤其在优化理论、计算数学、信号分析中有举足轻重的地位。因此，对正交矩阵性质的探讨是一个相当具有实际意义的课题。

本文首先从正交矩阵的等价定义入手研究正交矩阵的性质，通过对正交矩阵定义的理解，探讨正交矩阵经过一系列的矩阵运算，其结果是否还是正交矩阵，即是否保持正交性。再次，通过研究正交矩阵的特征值，以及正交矩阵的对角化，进一步加深对正交矩阵性质的理解，继而将正交矩阵与对角矩阵、正定矩阵相联系，更好的得出正交矩阵的一系列性质。最后，给出了正交矩阵在矩阵分解、矩阵对角化以及在空间坐标变换中的广泛应用。

关键字： 正交矩阵； 特征值； 对角化；对称矩阵；特征方程。

Abstract: Orthogonal matrix is a class of special matrix in the real number field, as an important branch of the matrix has a very wide range of applications in the natural sciences or social sciences, especially in the optimization theory, computational mathematics, signal analysis. Therefore, the nature of orthogonal matrix is a topic of considerable practical significance.

In order to study the nature of the orthogonal matrix, the paper will start from the equivalent definition of the orthogonal matrix. Based on the understanding definition of orthogonal matrix, explore the orthogonal matrix. After a series of matrix operations, the result is still orthogonal matrix, that is, whether to keep orthogonality. Study orthogonal matrix eigenvalue and orthogonal matrix diagonalization further deepen the understanding of the nature of the orthogonal matrix, then the orthogonal matrix with diagonal matrix, positive definite matrix associated with better draw orthogonal series of properties of the matrix.Finally through examples to illustrate the properties of orthogonal matrix in diagonal matrix, matrix decomposition, and widely used in space coordinate transformation.

Key words: Orthogonal matrix ; eigenvalues; diagonalization; special matrix; characteristic equation .