Solutions 11 - MASZlZ Linear Algebra I Exercises 11(with...

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Unformatted text preview: MASZlZ Linear Algebra I Exercises 11 (with solutions) © Francis J. Wright, 2006 Do not submit your solutions to this final coursework, which is not assessed and will not be marked. Solutions are available from the course web site. 1. For each of the following matrices A, find an orthogonal matrixP and a diagonal matrix A such that P A PT = A. 022 12 b 2—2 202 a , ,c . ()21()_25 () 220 VSolution (a) eigenvalues := —1, 3 —1 1 ei envectors := , g 1 1 anew Normalizing the eigenvectors gives PT = 1 1 and henceP= 35 35 L5 i5 2 2 _i5 ifi I 2 2 A is the diagonal matrix of eigenvalues 3 0 _ 0 —1 Check: in in ifi _ifi 2 2 2 2 1 0 PPT= . = _i5 L5 ifi i5 0 1 2 2 2 2 1 1 1 1 252512 2‘/7_2‘/7 30 PAPT: 1 1 ' 21 ' 1 1 = 0 1 ‘3” 35 3” 3J7 (b) eigenvalues := 1, 6 2 1 ’ eigenvectors := [ 1 %fi%5 Normalizing the eigenvectors gives PT = 1 2 and hence P = — J? — J? 5 5 2 J? i J? 5 5 —% J? %J? A is the diagonal matrix of eigenvalues 1 0 A: 0 6 Check: 35 iv? 15 Av? S 5 5 5 1 0 PPT: 1 2 ' 1 2 = 01 ——J? —¢? —¢? —¢? 5 5 5 5 2 1 2 1 T 5 J? 5 J? 2 —2 5 J? 5 J? l 0 PAP — . = -15 £5 ‘2 5 i5 £5 0 6 5 5 5 5 (C) eigenvalues := 4, —2, —2 1 —l —1 eigenvectors := l , l , 0 1 0 1 Call the eigenvectors x1, x2, x3. The last two eigenvalues are identical and the corresponding eigenvectors x2, x3 are linearly independent but not orthogonal, so replace the last eigenvector x3 by a linear combination of the last two eigenvectors x2, x3 that is orthogonal to )52 using x3 'x2 x3 := x3— x2 to give: x2 .x2 J? and hence P i 3 _L 6 0 5&2 J? 1_31_3 __ w e .W . g m n a m _ o _ 13 13 m a n 6 e m 1_31_21_6 g _ .m m E E 6 a m 1_31_21_6 O _ _ N A is the diagonal matrix of eigenvalues 0—20 400 00 A: —2 Check: 100 J? Hl01o 1 1 3 2 2 6 L 3 J? L 3 J? L 3 J? L 3 _L 6 1 322 i 3 001 0 J? i 3 J? PAPT 11 66JF3 _i 6 > Maple Solution 2. (a) Find an orthogonal matrix whose first row is the vector [ L, i J? J? 1 3 — — 10] and check it. m, J and check it. (b) Find an orthogonal matrix whose first column is the vector [ V Solution An orthogonal matrix has orthonormal rows and columns. (a) We need a vector in the direction orthogonal to the direction of the given vector, namely (1, 2). The orthogonal direction must be (2, —1) or (—2, 1). Normalizing this vector and making 1 2 1 2 it the second row of a matrix gives P1 = L or P2 = L . J? 2 —l J? —2 1 2 1 l 2 1 0 T 1 1 2 1 —2 1 0 Check:PPT=P2=— = ,PP =— = . 11152—1 01225—21 21 01 (b) We need a vector in the direction orthogonal to the direction of the given vector, namely (1, 3). The orthogonal direction must be (3, —1) or (—3, 1). Normalizing this vector and making 1 3 1 —3 1 orP2=L 3 —1 J10 J10 3 1 2 10 T1 1—3 13 10 _ ,P2P2=—[ :01 0 1 10 3 1 —3 1 3. LetA and B be symmetric matrices. Give a necessary and sufficient condition forA B to be symmetric. it the second column of a matrix gives P1 = 13 3—1 CheckzP1P1T=P12 = 11—0 > Maple Solution Solution IfA B is symmetric then (A B)T=A B. But (A B) T=BTAT =B A sinceA andB are symmetric. Hence, a necessary and sufficient condition forA B to be symmetric is that B A =A B, i.e. thatA and B are square matrices of the same size that commute. 4. Show that Q QT and QT Q are both symmetric matrices, where Q is any m x n matrix. Solution l(QQT)T=(QT)TQT=QQTand(QTQ)T=QT(QT)T=QTQ. 5. Prove that (A1A2---An)T =AZ:A5_1- "A; where A1, A2,. . ., A" are any matrices such that the productA1A2---An is defined. (State clearly any results you use.) Solution Suppose the result is true for n. Then 11(A1 A2- ”A" An +1)T =An +1 (A1 A2- -A n)T Tbecause (A B)T =BT AT as proved in Chapter 11 =An + 1 ATA_1--A1T by using the result for 11. Hence, the result is true for n + 1. It is clearly true for n = 1 so by induction it is true for all positive integer n. 6. (a) Let Q be any m X n real matrix andx any vector in IR". Prove that the scalar xT QT Qx Z 0. (b) LetZ be any m x n complex matrix andz be any vector in 0'. Let 2+ denote the transposed complex conjugate of 2, Le. 2*T, and || 2“ = \I z z. Prove that the real scalar z+Z+Zz 2 0 and that if Z is a square, non-singular n x 11 matrix andz at 0 then z+Z+ Z 2 > 0. Solution (a) Lety = Qx, which is a vector in R". ThenyTy = My || 2 and My || 2 2 0 by the norm axioms. Substituting the definition ofy gives 0 s yT y = ( Qx) T ( Qx) =xT QT Qx. (b) Lety =Zz, which is a vector in 0". Theny+y = My || 2 and My || 2 Z 0 by the norm axioms. Substituting the definition of y gives 0 S y+y = (Z z) + (Z z) =z+Z+Zz. If Z is a square, non- singular n x 11 matrix andz at 0 then y at 0, because if y = 0 thenz =Z_1 y => 2 = 0, which is a contradiction. If y at 0 then H y H 2 > 0 by the norm axioms, hence z+Z+ Z z > 0. ...
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