Convex Optimization - Inner Product

Inner product is a function which gives a scalar to a pair of vectors.

Inner Product − $f:mathbb{R}^n imes mathbb{R}^n ightarrow kappa$ where $kappa$ is a scalar.

The basic characteristics of inner product are as follows −

Let $X in mathbb{R}^n$

$left langle x,x ight anglegeq 0, forall x in X$

$left langle x,x ight angle=0Leftrightarrow x=0, forall x in X$

$left langle alpha x,y ight angle=alpha left langle x,y ight angle,forall alpha in kappa : and: forall x,y in X$

$left langle x+y,z ight angle =left langle x,z ight angle +left langle y,z ight angle, forall x,y,z in X$

$left langle overpne{y,x} ight angle=left ( x,y ight ), forall x, y in X$

Note −

Relationship between norm and inner product: $left | x ight |=sqrt{left ( x,x ight )}$

$forall x,y in mathbb{R}^n,left langle x,y ight angle=x_1y_1+x_2y_2+...+x_ny_n$

Examples

1. find the inner product of $x=left ( 1,2,1 ight ): and : y=left ( 3,-1,3 ight )$

Solution

$left langle x,y ight angle =x_1y_1+x_2y_2+x_3y_3$

$left langle x,y ight angle=left ( 1 imes3 ight )+left ( 2 imes-1 ight )+left ( 1 imes3 ight )$

$left langle x,y ight angle=3+left ( -2 ight )+3$

$left langle x,y ight angle=4$

2. If $x=left ( 4,9,1 ight ),y=left ( -3,5,1 ight )$ and $z=left ( 2,4,1 ight )$, find $left ( x+y,z ight )$

Solution

As we know, $left langle x+y,z ight angle=left langle x,z ight angle+left langle y,z ight angle$

$left langle x+y,z ight angle=left ( x_1z_1+x_2z_2+x_3z_3 ight )+left ( y_1z_1+y_2z_2+y_3z_3 ight )$

$left langle x+y,z ight angle=left { left ( 4 imes 2 ight )+left ( 9 imes 4 ight )+left ( 1 imes1 ight ) ight }+$

$left { left ( -3 imes2 ight )+left ( 5 imes4 ight )+left ( 1 imes 1 ight ) ight }$

$left langle x+y,z ight angle=left ( 8+36+1 ight )+left ( -6+20+1 ight )$

$left langle x+y,z ight angle=45+15$

$left langle x+y,z ight angle=60$