Inner Products 2.6 2002/07/24 2006/07/19 16:38:24.557 GMT-5 Michael Haag mjhaag@rice.edu Justin Romberg jrom@rice.edu Justin Romberg jrom@rice.edu Richard G. Baraniuk richb@rice.edu Michael Haag mjhaag@rice.edu Mariyah Poonawala mariyah@rice.edu Prashant Singh prash@ece.rice.edu Matthew Hutchinson mhutch@rice.edu dot product inner product inner products norm orthogonal projection This module describes the concept of inner products, which leads into our introduction of Hilbert spaces. Examples and properties of both of these concepts are discussed.

Definition: Inner Product You may have run across inner products, also called dot products, on n before in some of your math or science courses. If not, we define the inner product as follows, given we have some x n and y n inner product The inner product is defined mathematically as: x y y x y 0 y 1 … y n − 1 x 0 x 1 ⋮ x n − 1 i n 1 0 x i y i

Inner Product in 2-D If we have x 2 and y 2 , then we can write the inner product as x y x y θ where θ is the angle between x and y.

General plot of vectors and angle referred to in above equations.

Geometrically, the inner product tells us about the strength of x in the direction of y. For example, if x 1 , then x y y θ

Plot of two vectors from above example.

The following characteristics are revealed by the inner product: x y measures the length of the projection of y onto x. x y is maximum (for given x , y ) when x and y are in the same direction ( θ 0 θ 1 ). x y is zero when θ 0 θ 90° , i.e. x and y are orthogonal.

Inner Product Rules In general, an inner product on a complex vector space is just a function (taking two vectors and returning a complex number) that satisfies certain rules: Conjugate Symmetry: x y x y Scaling: α x y α x y Additivity: x y z x z y z "Positivity": x x 0 x x 0 orthogonal We say that x and y are orthogonal if: x y 0