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.)
Purchase answer to see full
attachment