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Unformatted text preview: 4. Orthogonal transformations and Rotations A matrix is defined to be orthogonal if the entries are real and (1) A A = I . Condition (1) says that the gram matrix of the sequence of vectors formed by the columns of A is the identity, so the columns are an orthonormal frame. An orthog- onal matrix defines an orthogonal transformation by mutiplying column vectors on the left. Condition (1) also shows that A is a rigid motion preserving angles and distances. A matrix satisfying (1) preserves the dot product; the dot product of two vectors is the same before and after an orthogonal transformation. This can be written as Av Aw = v w for all vectors v and w . This is true by the definition (1) of orthogonal matrix since Av Aw = ( Av ) Aw = v A Aw = v Iw = v w = v w . Thus lengths and angles are preserved, since they can be written in terms of dot products. The orthogonal transformation are a group since we can multiply two of them and get an orthogonal transformation. This is because if A and B are orthogonal, then A A = I and B B = I . So ( AB ) AB = B A AB = I , showing that AB is also orthogonal. Likewise we can take the inverse of an orthog- onal transformation to get an orthogonal transformation. Orthogonal transformations have determinant 1 or- 1 since by (1) and properties of determinant, (det A ) 2 = det( A )det A = det( A A ) = det I = 1 . 4.1. The rotation group. Orthogonal transformations with determinant 1 are called rotations, since they have a fixed axis. This is discussed in more detail below. The rotations also form a group. If we think of an orthogonal matrix A as a frame A = ( v 1 , v 2 , v 3 ) , then the determinant is the scalar triple product v 1 ( v 2 v 3 ) . The frame is right handed if the triple product is 1 and left handed if it is -1. The frame is the image of the right handed standard frame ( e 1 , e 2 , e 3 ) = I under the transformation A . Thus A preserves orientation (right-handedness) if the determinant is 1....
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This note was uploaded on 01/15/2012 for the course MAP 5485 taught by Professor Staff during the Fall '11 term at FSU.
- Fall '11