Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
5.3 Orthogonal Transformations and Orthogonal Matrices Definition 1 (5.3.1). A linear transformation T : Rn → Rn is called orthogonal if it preserves the length of vectors: ||T (x)|| = ||x|| , for all x ∈ Rn . If T (x) = Ax is an orthogonal transformation, we say that A is an orthogonal matrix. Theorem 2 (5.3.2, orthogonal transformations preserve orthogonality). Let T : Rn → Rn be an orthogonal linear transformation. If v, w ∈ Rn are orthogonal, then so are T (v), T (w). Note 3. In fact, orthogonal transformations preserve all angles, not just right angles: the angle between two nonzero vectors v, w ∈ Rn equals the angle between T (v), T (w). This is a homework problem. Theorem 4 (orthogonal transformations preserve the dot product). A linear transformation T : Rn → Rn is orthogonal if and only if T preserves the dot product: v · w = T (v) · T (w) for all v, w ∈ Rn . Theorem 5 (5.3.3, orthogonal matrices and orthonormal bases). An n × n matrix A is orthogonal if and only if its columns form an orthonormal basis of Rn . Theorem 6 (5.3.4, products and inverses of orthogonal matrices). a) The product AB of two orthogonal n × n matrices A and B is orthogonal. b) The inverse A−1 of an orthogonal n × n matrix A is orthogonal. Definition 7 (5.3.5). For an m × n matrix A, the transpose AT of A is the n × m matrix whose ijth entry is the jith entry of A: [AT ]ij = Aji . The rows of A become the columns of AT , and the columns of A become the rows of AT . A square matrix A is symmetric if AT = A and skew-symmetric if AT = −A. Note 8 (5.3.6). If v and w are two (column) vectors in Rn , then v · w = vT w. (Here we choose to ignore the difference between a scalar a and the 1 × 1 matrix a ). Theorem 9 (5.3.7, transpose criterion for orthogonal matrices). An n × n matrix A is orthogonal if and only if AT A = In or, equivalently, if A has inverse A−1 = AT . Theorem 10 (5.3.8, summary: orthogonal matrices). For an n × n matrix A, the following statements are equivalent: 1. A is an orthogonal matrix. 2. ||Ax|| = ||x|| for all x ∈ Rn . 1 3. The columns of A form an orthonormal basis of Rn . 4. AT A = In . 5. A−1 = AT . Theorem 11 (5.3.9, properties of the transpose). a) If A is an n × p matrix and B a p × m matrix (so that AB is defined), then (AB)T = B T AT . b) If an n × n matrix A is invertible, then so is AT , and (AT )−1 = (A−1 )T . c) For any matrix A, rank(A) = rank(AT ). − w1 .. Theorem 12 (invertibility criteria involving rows). For an n × n matrix A = . − wn following are equivalent: − , the − 1. A is invertible, 2. w1 , . . . , wn span Rn , 3. w1 , . . . , wn are linearly independent, 4. w1 , . . . , wn form a basis of Rn . Theorem 13 (column-row definition of matrix multiplication). Given matrices − w1 − | | .. A = v1 · · · vm and B = , . | | − wm − with v1 , . . . , vm , w1 , . . . , wm ∈ Rn , think of the vi as n × 1 matrices and the wi as 1 × n matrices. Then the product of A and B can be computed as a sum of m n × n matrices: AB = v1 w1 + · · · + vm wm = m X vi wi . i=1 Theorem 14 (5.3.10, matrix of an orthogonal projection). Let V be a subspace of Rn with orthonormal basis u1 , . . . , um . Then the matrix of the orthogonal projection onto V is | | QQT , where Q = u1 · · · um . | | 2