Question

3. Let B ERnxn be a symmetrie and P.D. matrix. Show that l s (o Bu) (B-v) for any nonzero v E R", and that the equality holds if and only if v is an eigenvector of B. (Hnt: note that llt -W/2t, B-1/2v), and use the Cauchy-Schwarz inequality.) 4. Let (ak) be a real sequence such that for each k, either akil > ak or akt? where, is a constant independent of k. Show that a 2 min(ai, T) for all k. (Hint: use induetion on k.)

Answer 1

3.Cauchy's Inequality

A special case of Hölder's sum inequality with modulus(v)4,2<=(v Transpose .B.V).(V Transpose B inverse .V where a k and B k are components equation Holds True V is an Eigen Vector of B.

k=1

(1)

where equality holds for $a_k=cb_k$. The inequality is sometimes also called Lagrange's inequality (Mitrinović 1970, p. 42), and can be written in vector form as

$||a·b||<=||a||||b||.$

(2)

In two-dimensions, it becomes

(a2 + b2) (c2 + d2) (ac+bd)2

(3)

It can be proven by writing

(a, x + b)-y|x + a ) =0.

(4)

If $b_i/a_i$ is a constant , then . If it is not a constant, then all terms cannot simultaneously vanish for real , so the solution is complex and can be found using the quadratic equation

$x=(-2suma_ib_i+/-sqrt(4(suma_ib_i)^2-4suma_i^2sumb_i^2))/(2suma_i^2).$

(5)

In order for this to be complex, it must be true that

$(sum_(i)a_ib_i)^2<=(sum_(i)a_i^2)(sum_(i)b_i^2),$

(6)

with equality when $b_i/a_i$ is a constant. The vector derivation is much simpler,

$(a·b)^2=a^2b^2cos^2theta<=a^2b^2,$

(7)

where

$a^2=a·a=sum_(i)a_i^2,$

(8)

and hence for V modulus 4 .2=(b Power 1/2v,B power -1/2v)