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We show that this is true if and only if $A=CB$ for some invertible matrix $C$ such that all entries of $C$ and $C^{-1}$ are bounded by some integer. We also show that every archimedean class contains a row echelon form and that its shape and archimedean classes (in $F$) of its pivots are uniquely determi"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1503.04346","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2015-03-14T20:40:26Z","cross_cats_sorted":[],"title_canon_sha256":"7fd77144df9a64c415fb58133d41fb6b2100f2c5f804e0396fa0712d9df9f4d3","abstract_canon_sha256":"e70517ef56071ec8aad80f0a189267d4bad196e01509d6aac5c4ab09db1ad571"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:17:55.559697Z","signature_b64":"gnC2wcGP2jHFBK3J9ChkdgIEWTiwUHwS30mTV4V4AaLmFWwy64qpNCK3BTLSDt/+qvS4AuipQ4MI9TXmVxBXBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"faffd11da997e059a55aee02cd1bdb59071a0a852a4b009ab968856ffbf7a171","last_reissued_at":"2026-05-18T00:17:55.559180Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:17:55.559180Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Archimedean classes of matrices over ordered fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RA","authors_text":"Jaka Cimpric","submitted_at":"2015-03-14T20:40:26Z","abstract_excerpt":"Let $(F,\\le)$ be an ordered field and let $A,B$ be square matrices over $F$ of the same size. 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