| 释义 |
Orthogonal MatrixAny Rotation can be given as a composition of rotations about three axes (Euler's Rotation Theorem), and thuscan be represented by a  Matrix operating on a Vector,
 | (1) |
In a Rotation, a Vector must keep its original length, so it must be true that
 | (2) |
 , 2, 3, where Einstein Summation is being used. Therefore, from the transformation equation,
 | (3) |
In order for this to hold, it must be true that
 | (5) |
 , 2, 3, where  is the Kronecker Delta. This is known as the Orthogonality Condition,and it guarantees that
 | (6) |
 | (7) |
 is the Matrix Transpose and I is the Identity Matrix. Equation (7) is theidentity which gives the orthogonal matrix its name. Orthogonal matrices have special properties which allow them to bemanipulated and identified with particular ease.
Let  and  be two orthogonal matrices. By the Orthogonality Condition, they satisfy
 | (8) |
 | (9) |
so the product  of two orthogonal matrices is also orthogonal.
The Eigenvalues of an orthogonal matrix must satisfy one of the following: - 1. All Eigenvalues are 1.
- 2. One Eigenvalue is 1 and the other two are
 . - 3. One Eigenvalue is 1 and the other two are Complex Conjugates of the form
 and  .
An orthogonal Matrix is classified as proper (corresponding to pure Rotation) if
 | (11) |
 is the Determinant of A,or improper (corresponding to inversion with possible rotation; Rotoinversion) if
 | (12) |
References
Arfken, G. ``Orthogonal Matrices.'' Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 191-205, 1985.Goldstein, H. ``Orthogonal Transformations.'' §4-2 in Classical Mechanics, 2nd ed. Reading, MA: Addison-Wesley, 132-137, 1980.
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