Analysis MOC

Matrix exponential

The matrix exponential uses the power series definition of the exponential function on matrices. Let be a real/complex matrix. Then is given by #m/def/anal/vec

This is convergent for all under any norm.

Proof of convergence

Let denote the Operator norm. Then

Since converges, the sequence converges absolutely and uniformly by the Weierstraß M-test. is finite-dimensional: All finite dimensional normed vector spaces are Banach and All norms on a finite dimensional space are equivalent. Thus the series converges, and does so regardless of norm.

Properties

For any , the following properties hold: #m/thm/anal/vec

  1. For any invertible , .
  2. uniquely solves with initial condition .
  3. if for all .
  6. for (see Pauli matrices)
Proof of properties 1–5

Let . Then

proving ^P1

Let

Then

and by the Existence and uniqueness theorem for IVPs this is unique, proving ^P2.

By Binomial expansion

proving ^P3

By basic properties of the Conjugate transpose

proving ^P4.

Let be the Jordan canonical form so . Then

proving ^P5

Generalisations

#state/develop | #lang/en | #SemBr