Inner and Outer Product of Matrix

Inner Product:

The inner product of two vectors u and v in the complex space, denoted as <uv> generates a complex as the output. The inner product must satisfy the following properties:

1. Linearity:

(<a|(w|b>+v|c>) = w<a|b>+v<a|c>

2. Symmetry:
(u|v) = (v|u)*

3. Positivity:
<u|u> ≥ 0 for |u> ≠ 0

Outer Product:

The outer product between u and v is written as:

|v> <u|