3 Cauchy Schwarz Inequality

last lecture in scope for MT1

Agenda

Cauchy Schwarz Inequality

folding in some signals

Cosine law for angles between elements in a vector space.

Complex EXP. (maybe periodicity)

Establish it in the simplest possible scenario to get a geometric inequality.

we can just get the intuition and move on. We have a Vector Space with elem x and y

CS inequality says:

|| <= ||x|| * ||y||

-||x|| * ||y|| <= <= ||x|| * ||y||

with equality iff x and y are parallel/ product is zero.

negative equality iff y = bx for some b<0 positive equality iff y = ax for some a<0

we are going to establish this in the simplist context.

Look at R^2 so we have x and y elements of R^2

im going to construct the normalized ersios of these two.

~x = x/||x|| , ~y = y/||y||

construct the vectors, then construct the normalized versions of these.

assume x has mag > 1

now lets define a new vector z which is ~x - ~y

start at y~ and point at x~

Draw an example in latex or ascii

the cauchy schwarz ineq is simply the statement that ||z|| >= 0

CS inequality is simply saying that the statement

0<= ||x||

0<= ||x||^2

0<= - - + = 2-2

i didnt pick complex calues, i ddint want o deal eomcplex conuusgation. multiple levels of integration to nring o ghr foad hofdign

if this is 2 it shoulf br less than equal ot we

2 <= 2 => = 1

= /||x||||y|| <= 1

<= ||x||||y||

nased on ehat we know so far. etc.

replace x with -x to get the other inqeuality.

what happens! negative x

< -x, y> <= ||-x||||y||