English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We show that the spectral theorem -- which we understand to be a statementthat every self-adjoint matrix admits a certain type of canonical form underunitary similarity -- admits analogues over other $*$-algebras distinct fromthe complex numbers. If these $*$-algebras contain nilpotents, then it is shownthat there is a consistent way in which many classic matrix decompositions --such as the Singular Value Decomposition, the Takagi decomposition, theskew-Takagi decomposition, and the Jordan decomposition, among others -- areimmediate consequences of these. If producing the relevant canonical form of aself-adjoint matrix were a subroutine in some programming language, then thecorresponding classic matrix decomposition would be a 1-line invocation with noadditional steps. We also suggest that by employing operator overloading in aprogramming language, a numerical algorithm for computing a unitarydiagonalisation of a complex self-adjoint matrix would generalise immediatelyto solving problems like SVD or Takagi. While algebras without nilpotents (likethe quaternions) allow for similar unifying behaviour, the classic matrixdecompositions which they unify are never obtained as easily. In the process ofdoing this, we develop some spectral theory over Clifford algebras of the form$\cl_{p,q,0}(\mathbb R)$ and $\cl_{p,q,1}(\mathbb R)$ where the former isadmittedly quite easy. We propose a broad conjecture about spectral theorems.

Unitary canonical forms over Clifford algebras, and an observed unification of some real-matrix decompositions