Manuscript XVII-A@1 · XVII-A.tex · lemma · 711d7ee4

Lemma: Sine perturbation bound

A conditional, reference-stable error budget · lines 754–771 · proof included or linked

XVII-A@1#lem:sine

For self-adjoint finite matrices $A, B$ ,

∥ sin A - sin B ∥_{\infty} \leq ∥ A - B ∥_{\infty} .

Proof. The Duhamel identity

e^{i A} - e^{i B} = i \int_{0}^{1} e^{i (1 - s) A} (A - B) e^{i s B} d s

gives $∥ e^{i A} - e^{i B} ∥_{\infty} \leq ∥ A - B ∥_{\infty}$ . Apply the same argument to $- A, - B$ and use $sin A = (e^{i A} - e^{- i A}) / (2 i)$ .

Displayed formulas in this passage

unlabeled · line 757
$∥ sin A - sin B ∥_{\infty} \leq ∥ A - B ∥_{\infty} .$
unlabeled · line 764
$e^{i A} - e^{i B} = i \int_{0}^{1} e^{i (1 - s) A} (A - B) e^{i s B} d s$

Nearby in this section

section: A conditional, reference-stable error budget
lemma: Lemma: Exact distance between binary record channels
proposition: Proposition: Named conditional perturbation allowance
section: A conditional, reference-stable error budget