Let the dual numbers be $\mathbb R[\varepsilon]/(\varepsilon^2)$. Write a general dual number as $a + b \varepsilon$ (where $\varepsilon^2 = 0$). Let $a + b \varepsilon \geq c + d \varepsilon$ mean that $a > c$ or $a = c$ and $b \geq d$. Given a symmetric matrix $M$ over the dual numbers (i.e. where $M = M^T$), it is known that $M$ can be written as $M = USU^T$ where $U^T = U^{-1}$ and $S$ is a unique diagonal dual-number matrix. The uniqueness is up to permutation of the diagonal entries, and subject to the constraint that all entries of $S$ are non-negative ($\geq 0$). [1]

It is known that the dual numbers have a relationship to analysis (especially differentiation). This is especially clear if you use big-oh notation. You then have that $(a + b \varepsilon)(c + d\varepsilon) = ac + \varepsilon (ad + bc) + O(\varepsilon^2)$, which is essentially dual number multiplication. Therefore there is an analysis version of the above theorem that can be expressed using O notation.

Is there an analysis-type proof of the above spectral theorem? I imagine that the $\varepsilon$ component of $S$ is a Gateaux differential.

[1] - Generalisations of Singular Value Decomposition to dual-numbered matrices. This is published in a journal, but the Arxiv version is better.