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 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: 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

• Vector flow
• Exponential map of Lie theory.

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