Introduction to Linear algebra-Strang-Solutions-Manual_ver13

31 a x t ax d ax t x d x t at x d t t problem set

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Unformatted text preview: 01 3 3 If A D 0 then all  D 0 so all  D 0 as in A D . If A is symmetric then 00 A3 D Qƒ3 QT D 0 requires ƒ D 0. The only symmetric A is Q 0 QT D zero matrix. 4  D 10 and 5 6 7 8 9 If  is complex then  is also an eigenvalue .Ax D x /. Always  C  is real. The 10 11 12 13 trace is real so the third eigenvalue of a 3 by 3 real matrix must be real. If x is not real then  D x T Ax =x T x is not always real. Can’t assume real eigenvectors! " # " #        1 1 1 1 31 9 12 :64 :48 :36 :48 2 2 2 2 D2 C4 1 1 ; D0 C25 1 1 13 12 16 :48 :36 :48 :64 2 2 2 2 "# xT 1 Œ x 1 x 2  is an orthogonal matrix so P1 C P2 D x 1 x T C x 2 x T D Œ x 1 x 2  D I; 1 2 xT 2 T T P1 P2 D x 1 .x1 x2 /x2 D 0. Second proof: P1 P2 D P1 .I P1 / D P1 P1 D 0 since 2 P1 D P1 .       0b A0 0A AD has  D i b and i b . The block matrices and are b0 0A A0 also skew-symmetric with  D i b (twice) and  D i b (twice). Solutions to Exercises 68 14 M is skew-symmetric and orthogonal; ’s must be i , i , 15 A D  i 1  i , i to have trace zero. 1 has  D 0; 0 and only one independent eigenvector x D .i; 1/. The i T good property for complex matrices is not AT D A (symmetric) but A D A (Hermitian with real eigenvalues and orthogonal eigenvectors: see Problem 20 and Section 10:2). 16 (a) If Az D y and AT y D z then BŒ y I z  D Œ AzI AT y  D Œ y I z . So  is also an eigenvalue of B . (b) AT Az D AT .y / D 2 z. (c)  D 1, 1, 1, 1; x 1 D .1; 0; 1; 0/, x 2 D .0; 1; 0; 1/, x 3 D .1; 0; 1; 0/, x 4 D .0; 1; 0; 1/. " # "# 001 1 p p 1, 17 The eigenvalues of B D 0 0 1 are 0; 2; 2 by Problem 16 with x 1 D 13 0 1 0 2 3 2 1 1 1 1 x 2 D 4 p 5, x 3 D 4 p 5. 2 2 18 1. y is in the nullspace of A and x is in the column space D row space because A D AT . Those spaces are perpendicular so y T x D 0. 2. If Ax D x and Ay D ˇ y then shift by ˇ : .A ˇI /x D . ˇ /x and .A ˇI /y D 0 and again x ?y . " # " # Perpendicular for A 1 10 10 1 1 0 ; B has S D 0 1 0 . Not perpendicular for B 19 A has S D 1 0 01 0 0 2d since B T ¤ B   T 1 3 C 4i 20 A D is a Hermitian matrix .A D A/. Its eigenvalues 6 and 4 are 3 4i 1 T real. Adjust equations .1/–.2/ in the text to prove that  is always real when A D A: T Ax D x leads to Ax D x : Transpose to x T A D x T  using A D A: Then x T Ax D x T x and also x T Ax D x T x : So  D  is real:  12 21 (a) False. A D 01  (b) True from AT D QƒQT (c) True from A 1 D Qƒ 1 QT (d) False!  01 has 10 1 D i and 2 D i with x 1 D .1; i / first for A but x 1 D .1; i / first for AT . 22 A and AT have the same ’s but the order of the x ’s can change. A D  23 A is invertible, orthogonal, permutation, diagonalizable, Markov; B is projection, di- agonalizable, Markov. A allows QR; SƒS 1 ; QƒQT ; B allows SƒS 24 Symmetry gives QƒQT if b D 1; repeated  and no S if b D 1 and QƒQT . 1; singular if b D 0. 25 Orthogonal and symmetric requires jj D    real, so   ˙1. Then A D ˙I  1 and D or  A D QƒQT D cos  sin...
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This note was uploaded on 09/25/2012 for the course PHY 103 taught by Professor Minki during the Spring '12 term at Korea Advanced Institute of Science and Technology.

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