Numerical Mathematics Problems

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Mathematics

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3.16 Let A be an n x n matrix. The trace of A, denoted by tr (A), is defined by tr (A) = Σ air- i=1 Show that tr (ATA) = || A|| 3. Also show tr (A) = "hi, and det A = 77" = 1 dis where the h; are the eigenvalues of A. (Refer to the coefficients of the characteristic polynomial and the fact that the li are the roots of det (2.1 - A).) Show that if B U-1AU, then tr (B) tr (A). Further, if U is a unitary matrix. then tr (B*B) = tr(A* A), that is, 71 || B* B|| 3 = || A* A|| Σ |b? . į lail? or i.j = 1 i. = 1 25 a) Let А (a11 412 021 022/ with a 12 = 221. Let U be the rotation matrix cos sin e sin COS os e) with tan 20 = 2012/(a22 - 0u), – 7/4 so s T/4. Verify that B = U*AU is a diagonal matrix. Use Exercise 3.16 to prove that bı + 632 = aſi + 2aíz + ažz. Verify that U* = U-1. b) Let A = (dij) be an n x n real symmetric matrix. Let U = (uj) be a two-dimensional rotation matrix defined by and Hij = = bij Urr = uss = cos 0, - Usr sin 0, uu = 1, i #r, s 0 otherwise. Show that U* U-1. Let B UAU* Verify that for i #r, s and j * r, s, dij, whereas in the r and s rows and columns, we have = dir cos 0 + ais sin 0, ir, S -air sin @ + dis cos 0, = arr cos? 0 + 2ars cos 0 sin @ + ags sin2 0, bgs = Arr sin? 0) – 2drs cos 0 sin 0 + ass cos? 0, (ass - arr) sin 0 cos 0 + ars (cosa ( – sin? 0). bri bir bis bsi } = brr bsr = brs [Hint: Show that the multiplication by U affects only the r and s rows and the multiplica- tion by U* affects only the r and s columns.] Show that a rotation through angle given by tan 20 2ars/(arr - ass), 0 s 101 s 1/4, makes bys = 0. c) Referring to part (b), show that for i # r, s, bir + b = a + as and (b + b}) Σ (a + a?). = iris irs Similarly, Σ (b) + b3) (am + a3). i #ris iuris Show that, if brs = bsr = 0, Σβή Şbi = £, aš – 2a. (Recall that dij bij if neither index i, jis r or s.)
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