Let’s discuss an important theorem.
Sylvester’s Law of Inertia
We will define sesquilinear forms, the property Hermitian, and the associated quadratic form.The nice thing is that the polarization identity always holds, which shows that the sesquilinear form $s : V \times V \rightarrow \mathbb{F}$ is completely determined by its associated quadratic form $q : V \rightarrow \mathbb{R}$. Moreover, we will show that any quadratic form can be represented by a self-adjoint matrix and we can even restrict ourselves to diagonal matrices with only $1$, $-1$, and $0$ as entries. Finally, we will show that the occurrence of these number is invariant. We can speak of the positive inertia and negative inertia and the difference is called signature of the quadratic form.
Content of the video:
00:00 Introduction
00:30 Example of quadratic form
01:30 Definition (sesquilinear form)
04:25 Definition quadratic form
05:15 Polarization identity
06:40 Matrix representation of quadratic form
09:58 Diagonalization of quadratic form
11:30 Proof of diagonalization
18:36 Decomposition of vector space
21:00 Theorem: Sylvester’s law of inertia
22:12 Proof of theorem
24:59 Credits