LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generated

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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Special unitary groups are connected

Statement

For every integer $n \geq 1$ , the matrix group $S U (n) = {A \in M_{n} (C) : A^{*} A = I, \det A = 1}$ is path connected.

Facts & Assumptions

Given: An integer $n \geq 1$ and a matrix $A \in S U (n)$ .

[F1]

A normal complex matrix has an orthonormal eigenbasis. Complex spectral theorem: a normal endomorphism of a finite-dimensional complex inner product space has an orthonormal eigenbasis, and conversely

[F2]

For real $α$ , $∣ e^{i α} ∣ = 1$ , and $e^{i α} = e^{i β}$ exactly when $α - β \in 2 π Z$ . $\exp (x + i y) = e^{x} (\cos y + i \sin y)$ , $∣ \exp (x + i y) ∣ = e^{x}$ , and $e^{i π} + 1 = 0$ $\ker (\exp) = 2 π i Z$ , and $\exp z = \exp w$ exactly when $z - w \in 2 π i Z$

[F3]

The complex exponential is entire, hence continuous. The complex exponential is entire and its complex derivative is itself

Proof

1.1givenF1F2choosealgebra

The matrix $A$ is unitary and therefore normal. By [F1], choose a unitary matrix $Q$ and real phases $α_{1}, \dots, α_{n}$ such that $A = Q diag (e^{i α_{1}}, \dots, e^{i α_{n}}) Q^{*}$ . Since $\det A = 1$ , [F2] gives $\sum_{j} α_{j} = 2 π k$ for some integer $k$ . Replace $α_{1}$ by $α_{1} - 2 π k$ ; the eigenvalue is unchanged and the sum of phases is now zero. This uses only finitely many eigenvalues.

2.1givenF2F3step 1.1algebra∎

Define $A (t) = Q diag (e^{i t α_{1}}, \dots, e^{i t α_{n}}) Q^{*}$ for $0 \leq t \leq 1$ . Each $A (t)$ is unitary by [F2], and $\det A (t) = e^{i t \sum_{j} α_{j}} = 1$ , so $A (t) \in S U (n)$ . The path is continuous by [F3], $A (0) = I$ , and $A (1) = A$ . Thus every element is joined to the identity, proving path connectedness.

Depends on

Used by

Dependency tree · two levels

25 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Chetan Nayak, Steven H. Simon, Ady Stern, Michael Freedman, and Sankar Das Sarma, Non-Abelian Anyons and Topological Quantum Computation (standard reference, not scraped)
Michael H. Freedman, Michael J. Larsen, and Zhenghan Wang, A Modular Functor Which Is Universal for Quantum Computation (standard reference, not scraped)