Ch 5: ORTHOGONALITY 5.5 Orthonormal Sets 1. a set fv1; v2; : : : ; vng is an orthogonal set if hvi; vji = 0, 81 ≤ i 6= j; ≤ n (note that orthogonality only makes sense in an inner product space since you need to inner product to check hvi; vji = 0. n For example fe1; e2; : : : ; eng is an orthogonal set in R . 2. fv1; v2; : : : ; vng is an orthogonal set =) v1; v2; : : : ; vn are linearly independent. 3. a set fv1; v2; : : : ; vng is an orthonormal set if hvi; vji = 0 and jjvijj = 1, 81 ≤ i 6= j; ≤ n (i.e. orthogonal and unit length). n For example fe1; e2; : : : ; eng is an orthonormal set in R . 4. given an orthogonal basis for a vector space V , we can always find an orthonormal basis for V by dividing each vector by its length (see Example 2 and 3 page 256) n 5. a space with an orthonormal basis behaves like the euclidean space R with the standard basis (it is easier to work with basis that are orthonormal, just like it is n easier to work with the standard basis versus other bases for R ) 6. if v is a vector in a inner product space V with fu1; u2; : : : ; ung an orthonormal basis, Pn Pn then we can write v = i=1 ciui = i=1hv; uiiui Pn 7. an easy way to find the inner product of two vectors: if u = i=1 aiui and v = Pn i=1 biui, where fu1; u2; : : : ; ung is an orthonormal basis for an inner product space Pn V , then hu; vi = i=1 aibi Pn 8. Parseval's Formula: if u = i=1 aiui, where fu1; u2; : : : ; ung is an orthonormal basis Pn 2 Pn 2 for an inner product space V , then jjujj = hu; ui = i=1 ci = i=1hu; uii (where ci = hv; uii by Thm 5.5.2) 1 9. an n × n matrix Q is said to be an orthogonal matrix if the column vectors of Q are orthonormal. (Note that terminology does not imply that the vectors have length one, but the definition says they must be.) 10. Q is orthogonal () QT Q = I (or QQT = I) 11. properties of orthogonal matrices Q: (a) the columns of Q form an orthonormal basis (b) QT Q = I = QQT (c) Q−1 = QT (is very easy to compute Q−1 for orthogonal matrices) (d) hQx;Qyi = hx; yi (multiplication by an orthogonal matrix preserves the angle between two vectors) (e) jjQxjj2 = jjxjj2 (multiplication by an orthogonal matrix preserves length of vec- tors) 12. so all that an orthogonal matrix does to vectors is to shift them around and rotate all the vectors by the same amount (it is an ideal linear transformation) 13. how do we solve Ax = y for x if A is an orthogonal matrix? Ax = y AT Ax = AT y x = AT y 14. If A is not orthogonal, then A can be decomposed as QR-factorization, where Q is orthogonal, and R is up upper triangular matrix, since orthogonal matrices are easy to work with, and the upper triangular ones are easy to use for solving (back substitution){see next Section 5.6. Another alternative when A is not orthogonal is SVD-decomposition (Section 6.5) 2 15. a permutation matrix is a matrix obtained from I be reordering its columns (it is an elementary matrix of type I, i.e. obtained by swapping two rows). 16. permutation matrices are orthogonal matrices (and so P −1 = P T ), and P −1 is also an orthogonal matrix 17. if I3 has columns e1; e2; e3, and P is the permutation matrix that reorders the columns 20 1 03 of I3 to e2; e1; e3, then we say that P = [e2; e1; e3] = 41 0 05. 0 0 1 2 3 b1 18. for an n × 3 matrix A = [a1; a2; a3] and a 3 × n matrix B = 4b25 we have b3 AP = [Ae2;Ae1;Ae3] = [a2; a1; a3] and 2 3 b2 T T T PB = [e2 B; e1 B; e3 B] = 4b15 b3 19. feel free to read the end of the section, I will not be covering it in class. 3.