“orthogonal matrix”的概念、定义、习题解题方法及翻译-数学辞典

单词

orthogonal matrix

释义

orthogonal matrix

A square matrix A is orthogonal if A^TA = I, where A^T is the transpose of A. The following properties hold:

(i) If A is orthogonal, A−1 = A^T and so AA^T = I.

(ii) If A is orthogonal, det A = ±1.

(iii) If A and B are orthogonal matrices of the same order, then AB is orthogonal, as is A−1.

The orthogonal n × n matrices form a group O(n), and those with determinant 1 form a subgroup SO(n). The orthogonal matrices are the linear (see linear map) isometries of Rn.

2 × 2 orthogonal matrices have the form

The first matrix represents rotation by θ anticlockwise about the origin. The second represents reflection in the line y = xtan(θ/2). See unitary matrix.

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