Question
Transcribed Text
(1) [5 points] For x, y € C", recall that
n
=
(1)
k=1
If x, y, 2 E Cn and a, B €.C, prove that (ax + By,z) = a(x,2) + B(y,2).
(2) [5 points] For x E C", recall that
n
==
(2)
k=1
If x € Cn and a € C, use properties of the modulus to prove that
(3) [5 points] If p(z) = 22 + a12 + ao € P2(C) and
C: 10-11 0 ao - 1 ai
,
prove that pc = p.
(4) [10 points] A matrix A E Mn (C) is called negative definite if 2* Az < 0, Vz € Cn, 2 # 0. If A
is negative definite and \ € spec(A), prove that \ < 0. Hint: Compute U* Av, in which U is an
eigenvector of A associated with 1.
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