Introduction to Linear algebra-Strang-Solutions-Manual_ver13

Key point elimination from both sides gives the

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Unformatted text preview: d b2 e bc The examples 4 3 9 and b d e lead to and . 7 32 e bc f c2 890 cef " # " #" #" # " #" # 1 1 1 0 1 1 1 1 2 0 1 1; 1 AD 1 1 1 1 22 1 AD 0 1 1 231 1 1 201 1 2 3 0001 This cyclic P exchanges rows 1-2 then 61 0 0 07 23 A D 4 D P and L D U D I . 0 1 0 05 rows 2-3 then rows 3-4. 0010 " #" #" #" # 1 012 1 2 1 1 1 038D0 1 3 8 . If we wait 24 PA D LU is 1 211 0 1=3 1 2 =3 " #" #" # 1 1 211 1 0 1 2. to exchange and a12 is the pivot, A D L1 P1 U1 D 3 1 11 002 25 The splu code will not end when abs.A.k; k// < tol line 4 of the slu code on page 100. Instead splu looks for a nonzero entry below the diagonal in the current column k , and executes a row exchange. The 4 lines to exchange row k with row r are at the end of Section 2.7 (page 113). To find that nonzero entry A.r; k/, follow abs.A.k; k// < tol by locating the first nonzero (or the largest A.r; k/ out of r D k C 1; : : : ; n). 26 One way to decide even vs. odd is to count all pairs that P has in the wrong order. Then P is even or odd when that count is even or odd. Hard step: Show that an exchange always switches that count! Then 3 or 5 exchanges will leave that count odd. # # " " 1 100 T 31 puts 0 in the 2; 1 entry of E21 A. Then E21 AE21 D 0 2 4 27 (a) E21 D 049 1 # " 1 1 is still symmetric, with zero also in its 1, 2 entry. (b) Now use E32 D 41 T T to make the 3, 2 entry zero and E32 E21 AE21 E32 D D also has zero in its 2, 3 entry. Key point: Elimination from both sides gives the symmetric LDLT directly. 2 3 0123 61 2 3 07 28 A D 4 D AT has 0; 1; 2; 3 in every row. (I don’t know any rules for a 2 3 0 15 3012 symmetric construction like this) Solutions to Exercises 26 29 Reordering the rows and/or the columns of a b c d will move the entry a. So the result cannot be the transpose (which doesn’t move a). " #" # " # 1 0 1 yBC yBC C yBS 1 1 0 yCS yBC C yCS . 30 (a) Total currents are AT y D D 0 1 1 yBS yCS yBS (b) Either way .Ax /T y D x T .AT y / D xB yBC C xB yBS xC yBC C xC yCS xS yCS xS yBS . " #   " 700 #   1 50   x1 1 40 2 6820 1 truck T 3 31 40 1000 D Ax ; A y D D x2 50 1000 50 188000 1 plane 2 50 3000 32 Ax  y is the cost of inputs while x  AT y is the value of outputs. 33 P 3 D I so three rotations for 360ı ; P rotates around .1; 1; 1/ by 120ı . 34  1 4   2 10 1 D 9 21 2 35 L.U T /  2 D EH D (elementory matrix) times (symmetric matrix). 5 1 is lower triangular times lower triangular, so lower triangular. The transpose of U DU is U T D T U T T D U T DU again, so U T DU is symmetric. The factorization multiplies lower triangular by symmetric to get LDU which is A. T 36 These are groups: Lower triangular with diagonal 1’s, diagonal invertible D , permuta1 tions P , orthogonal matrices with QT D Q .  1  1 is southeast: 1 1 D 0 1. 10 1 The rows of B are in reverse order from a lower triangular L, so B D PL. Then B 1 D L 1 P 1 has the columns in reverse ord...
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