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Given tuples $X=(X_1, \\dots, X_g) \\in SM_{n_1}(\\mathbb{R})^g$ and $Y=(Y_1, \\dots, Y_g) \\in SM_{n_2}(\\mathbb{R})^g$, a matrix convex combination of $X$ and $Y$ is a sum of the form \\[ V_1^* XV_1+V_2^* Y V_2 \\quad \\quad \\quad V_1^* V_1+V_2^* V_2=I_n \\] where $V_1:\\mathbb{R}^n \\to \\mathbb{R}^{n_1}$ and $V_2:\\mathbb{R}^n \\to \\mathbb{R}^{n_2}$ are contractions. Matrix convex sets are sets which are closed under matrix convex combinations. A key feature of matrix convex combinations is that the $g$-tuples $X, Y$, an"},"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":"1806.09053","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2018-06-23T23:42:44Z","cross_cats_sorted":[],"title_canon_sha256":"f74edeba503fbae2a2147146b7f2726a9babcb4fbb5a369f37c6da7e1ab83252","abstract_canon_sha256":"973f1864a242f704bb3dd25e749c494ae22697397b53d65cf3e6f11d5ac76c5b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T03:59:19.784154Z","signature_b64":"PvWK9q774zZ8VffOJaL6eJPxZ9gCVA/aNf+qqS7IICPuMXUqDtQQ7C0X4ySCB9ywjQWU4NtJd3vhU2HbyionBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a19dcdb4ab1528f9087b061bcbeeea779197b0ea3e825e295dd31d4a547701fe","last_reissued_at":"2026-07-05T03:59:19.783647Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T03:59:19.783647Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Arveson extreme points span free spectrahedra","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.OA","authors_text":"Eric Evert, J. William Helton","submitted_at":"2018-06-23T23:42:44Z","abstract_excerpt":"Let $ SM_n(\\mathbb{R})^g$ denote $g$-tuples of $n \\times n$ real symmetric matrices. Given tuples $X=(X_1, \\dots, X_g) \\in SM_{n_1}(\\mathbb{R})^g$ and $Y=(Y_1, \\dots, Y_g) \\in SM_{n_2}(\\mathbb{R})^g$, a matrix convex combination of $X$ and $Y$ is a sum of the form \\[ V_1^* XV_1+V_2^* Y V_2 \\quad \\quad \\quad V_1^* V_1+V_2^* V_2=I_n \\] where $V_1:\\mathbb{R}^n \\to \\mathbb{R}^{n_1}$ and $V_2:\\mathbb{R}^n \\to \\mathbb{R}^{n_2}$ are contractions. Matrix convex sets are sets which are closed under matrix convex combinations. 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