Matrix exponential

The matrix exponential $e x p : ℂ 𝑛 \times 𝑛 \to ℂ 𝑛 \times 𝑛$ uses the power series definition of the exponential function on matrices. Let $𝐀$ be a $𝑛 \times 𝑛$ real/complex matrix. Then $𝑒 𝐀$ is given by #m/def/anal/vec

𝑒 𝐀 = \infty \sum 𝑘 = 0 𝐀 𝑘 𝑘!

This is convergent for all $𝐀 \in ℂ 𝑛 \times 𝑛$ under any norm.

Proof of convergence

Let $‖ \cdot ‖$ denote the Operator norm. Then

0 \leq ‖ 𝐀 𝑘 𝑘! ‖ = ‖ 𝐀 𝑘 ‖ 𝑘! \leq ‖ 𝐀 ‖ 𝑘 𝑘!

Since $𝑒 ‖ 𝐴 ‖ = \sum \infty 𝑘 = 0 ‖ 𝐴 ‖ 𝑘 𝑘!$ converges, the sequence converges absolutely and uniformly by the Weierstraß M-test. $ℂ 𝑛 \times 𝑛$ 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 $𝐀 \in ℂ 𝑛 \times 𝑛$ , the following properties hold: #m/thm/anal/vec

For any invertible $𝐓 \in G L (𝑛)$ , $𝑒 𝐓 𝐀 𝐓 - 1 = 𝐓 𝑒 𝐀 𝐓 - 1$ .
$𝑒 𝑡 𝐀$ uniquely solves $˙ 𝐗 (𝑡) = 𝐀 𝐗 (𝑡)$ with initial condition $𝐗 (0) = 𝐈$ .
$𝑒 𝑡 𝐀 𝑒 𝑠 𝐁 = 𝑒 𝑡 𝐀 + 𝑠 𝐁$ if $𝐀 𝐁 = 𝐁 𝐀$ for all $𝑡, 𝑠 \in ℂ$ .
$(𝑒 𝐀) † = 𝑒 (𝐀 †)$
$d e t 𝑒 𝐀 = 𝑒 t r 𝐀$
$𝑒 - 𝑖 𝜙 ⃗ 𝜎 \cdot ⃗ 𝐧 / 2 = c o s (𝜙 / 2) 𝐈 - 𝑖 s i n (𝜙 / 2) ⃗ 𝜎 \cdot ⃗ 𝐧$ for $⃗ 𝐧 \in 𝕊 3$ (see Pauli matrices)

Proof of properties 1–5

Let $𝐓 \in G L (n)$ . Then

𝑒 𝐓 𝐀 𝐓 - 1 = \infty \sum 𝑘 = 0 (𝐓 𝐀 𝐓 - 1) 𝑘 𝑘! = \infty \sum 𝑘 = 0 𝐓 𝐀 𝑘 𝐓 - 𝟏 𝑘! = 𝐓 𝑒 𝐀 𝐓 - 1

proving ^P1

Let

𝐗 (𝑡) = 𝑒 𝑡 𝐀 = \infty \sum 𝑘 = 0 (𝑡 𝐀) 𝑘 𝑘!

Then

˙ 𝐗 (𝑡) = \infty \sum 𝑘 = 1 𝑡 𝑘 - 1 𝐀 𝑘 (𝑘 - 1)! = 𝐀 \infty \sum 𝑘 = 0 (𝑡 𝐀) 𝑘 𝑘! = 𝐀 𝐗 (𝑡)

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

By Binomial expansion

𝑒 𝑡 𝐀 𝑒 𝑠 𝐁 = (\infty \sum 𝑘 = 0 (𝑡 𝐀) 𝑘 𝑘!) (\infty \sum ℓ = 0 (𝑠 𝐁) ℓ ℓ!) = \infty \sum 𝑘 = 0 𝑘 \sum ℓ = 0 𝑡 ℓ 𝑠 𝑘 - ℓ 𝐀 ℓ 𝐁 𝑘 - ℓ ℓ! (𝑘 - ℓ)! = \infty \sum 𝑘 = 0 𝑘 \sum ℓ = 0 𝑘! ℓ! (𝑘 - ℓ)! 𝑡 ℓ 𝑠 𝑘 - ℓ 𝐀 ℓ 𝐁 ℓ - 𝑘 𝑘! = \infty \sum 𝑘 = 0 𝑘 \sum ℓ = 0 (𝑘 ℓ) 𝑡 ℓ 𝑠 𝑘 - ℓ 𝐀 ℓ 𝐁 ℓ - 𝑘 𝑘! = \infty \sum 𝑘 = 0 (𝑡 𝐀 + 𝑠 𝐁) 𝑘 𝑘! = 𝑒 (𝑡 𝐀 + 𝑠 𝐁)

proving ^P3

By basic properties of the Conjugate transpose

(𝑒 𝐀) † = (\infty \sum 𝑘 = 0 𝐀 𝑘 𝑘!) † = \infty \sum 𝑘 = 0 (𝐀 𝑘) † 𝑘! = \infty \sum 𝑘 = 0 (𝐀 †) 𝑘 𝑘! = 𝑒 (𝐀 †)

proving ^P4.

Let $𝐃$ be the Jordan canonical form so $𝐀 = 𝐒 𝐃 𝐒 - 1$ . Then

d e t 𝑒 𝐀 = d e t 𝑒 𝐃 = 𝑛 \prod 𝑘 = 1 𝑒 𝐷 𝑘 𝑘 = 𝑒 t r 𝐃 = 𝑒 t r 𝐀

proving ^P5

Generalisations

Vector flow
Exponential map of Lie theory.

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