Convex Optimization - Inner Product

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

Inner Product − $f:\mathbb{R}^n \times \mathbb{R}^n\rightarrow \kappa$ where $\kappa$ is a scalar.

The basic characteristics of inner product are as follows −

Let $X \in \mathbb{R}^n$

• $\left \langle x,x \right \rangle\geq 0, \forall x \in X$

• $\left \langle x,x \right \rangle=0\Leftrightarrow x=0, \forall x \in X$

• $\left \langle \alpha x,y \right \rangle=\alpha \left \langle x,y \right \rangle,\forall \alpha \in \kappa \: and\: \forall x,y \in X$

• $\left \langle x+y,z \right \rangle =\left \langle x,z \right \rangle +\left \langle y,z \right \rangle, \forall x,y,z \in X$

• $\left \langle \overline{y,x} \right \rangle=\left ( x,y \right ), \forall x, y \in X$

Note −

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

• $\forall x,y \in \mathbb{R}^n,\left \langle x,y \right \rangle=x_1y_1+x_2y_2+...+x_ny_n$

Examples

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

Solution

$\left \langle x,y \right \rangle =x_1y_1+x_2y_2+x_3y_3$

$\left \langle x,y \right \rangle=\left ( 1\times3 \right )+\left ( 2\times-1 \right )+\left ( 1\times3 \right )$

$\left \langle x,y \right \rangle=3+\left ( -2 \right )+3$

$\left \langle x,y \right \rangle=4$

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

Solution

As we know, $\left \langle x+y,z \right \rangle=\left \langle x,z \right \rangle+\left \langle y,z \right \rangle$

$\left \langle x+y,z \right \rangle=\left ( x_1z_1+x_2z_2+x_3z_3 \right )+\left ( y_1z_1+y_2z_2+y_3z_3 \right )$

$\left \langle x+y,z \right \rangle=\left \{ \left ( 4\times 2 \right )+\left ( 9\times 4 \right )+\left ( 1\times1 \right ) \right \}+$

$\left \{ \left ( -3\times2 \right )+\left ( 5\times4 \right )+\left ( 1\times 1\right ) \right \}$

$\left \langle x+y,z \right \rangle=\left ( 8+36+1 \right )+\left ( -6+20+1 \right )$

$\left \langle x+y,z \right \rangle=45+15$

$\left \langle x+y,z \right \rangle=60$