need to know insight concept or underlaying concept of Cauchy–Schwarz inequality

CAN ANYONE PLEASE TELL ME ABOUT

Cauchy–Schwarz inequality

1. WHAT IT MEANS ACTUALLY I KNOW FORMAL DEFINITION BUT I WANA INSIGHT CONCEPT WHY PRODUCT OF INDIVIDUAL VECTORS IS ALWAYS> AND EQUAL THAN INNER PRODUCT

2. SPECIFY THE CASE WHEN PRODUCT OF MAGNITUDES OF IND.VECTORS IS = MAG OF INNER PRODUCT

MAY THIS HELPS ME TO GET INSIGHT CONCEPT

I AM NEW TO THESES FORUMS IF ANYONE CAN EMAIL ME IT WILL BE GREAT FOR ME...

THANKX.

E-MAIL ADRESS:

junaidanwar194@gmail.com

Re: need to know insight concept or underlaying concept of Cauchy–Schwarz inequality

Suppose you have two UNIT vectors $\displaystyle \hat{v}, \hat{w}$ in some inner product space (V, <,>).

Then $\displaystyle \vec{z} = <\hat{v}, \hat{w}> \hat{w}$ is the component of $\displaystyle \hat{v}$ in the $\displaystyle \hat{w}$ direction.

(Recall that $\displaystyle \hat{v} = \vec{z} + (\hat{v} - \vec{z})$, decomposes $\displaystyle \hat{v}$ into parallel and perpendicular components w/resp to $\displaystyle \hat{w}$:

$\displaystyle <(\hat{v} - \vec{z}), \hat{w}> = <\hat{v}, \hat{w}> - <\vec{z}, \hat{w}> = <\hat{v}, \hat{w}> - <<\hat{v}, \hat{w}> \hat{w}, \hat{w}>$

$\displaystyle = <\hat{v}, \hat{w}> - <\hat{v}, \hat{w}><\hat{w}, \hat{w}> = <\hat{v}, \hat{w}>( 1 - <\hat{w}, \hat{w}>) = 0$.)

__Conceptually__, should $\displaystyle \hat{v}$, some generic unit vector, have a greater component in the $\displaystyle \hat{w}$ direction than $\displaystyle \hat{w}$ itself does?

No. __Conceptually__, each unit vector "should have the greatest magnitude among all unit vector in its own direction." Right?

OK. So that means we __expect__.... $\displaystyle \lVert \vec{z} \rVert \le \lVert \hat{w} \rVert = 1$, with equality iff $\displaystyle \hat{v} = \pm \hat{w}$.

Thus we __expect__.... $\displaystyle \lVert <\hat{v}, \hat{w}> \hat{w} \rVert \le 1$, with equality iff $\displaystyle \hat{v} = \pm \hat{w}$.

Thus we __expect__.... $\displaystyle \lvert <\hat{v}, \hat{w}> \rvert \lVert \hat{w} \rVert \le 1$, with equality iff $\displaystyle \hat{v} = \pm \hat{w}$.

Thus we __expect__.... $\displaystyle \lvert <\hat{v}, \hat{w}> \rvert \le 1$, with equality iff $\displaystyle \hat{v} = \pm \hat{w}$.

Thus we __expect__ inner products of *unit* vectors to take values in [-1,1], with it being $\displaystyle \pm1$ only when one vector is $\displaystyle \pm1$ the other.

Now, suppose $\displaystyle \vec{a}$ and $\displaystyle \vec{b}$ are any two non-zero vectors (CS obviously holds if one of the vectors is $\displaystyle \vec{0}$). What do we expect?

Consider the proceeding in light of their unit vectors $\displaystyle \frac{\vec{a}}{\lVert \vec{a} \rVert}$ and $\displaystyle \frac{\vec{b}}{\lVert \vec{b} \rVert}$:

We __expect__.... $\displaystyle \left\lvert \left<\frac{\vec{a}}{\lVert \vec{a} \rVert}, \frac{\vec{b}}{\lVert \vec{b} \rVert}\right> \right\rvert \le 1$, with equality iff $\displaystyle \frac{\vec{a}}{\lVert \vec{a} \rVert} = \pm \frac{\vec{b}}{\lVert \vec{b} \rVert}$, i.e. iff $\displaystyle \vec{a} = \left( \pm \frac{\lVert \vec{a} \rVert }{\lVert \vec{b} \rVert } \right)\vec{b}$.

So we __expect__.... $\displaystyle \lvert <\vec{a}, \vec{b}> \rvert \le \lVert \vec{a} \rVert \ \lVert \vec{b} \rVert$, with equality iff $\displaystyle \vec{a}$ = some multiple of $\displaystyle \vec{b}$.

Thus we __expect__ the Cauchy-Schwarz inequality to hold - because we expect inner products of unit vectors to take values in [-1,1].

That's its conceptual meaning to me.