Pith Number

pith:RYKQN7PK

pith:2026:RYKQN7PKB7IFORF6AXYRKDLQL6

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Block Krylov subspaces and orthogonal matrix polynomials: a structural correspondence with applications to unitary matrices

Michele Rinelli, Raf Vandebril

Polynomial block Krylov subspaces are isometrically isomorphic to spaces of matrix polynomials of bounded degree under a no-deflation assumption.

arxiv:2605.16954 v1 · 2026-05-16 · math.NA · cs.NA

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Claims

C1strongest claim

Under a no-deflation assumption, polynomial block Krylov subspaces are isometrically isomorphic to spaces of matrix polynomials of bounded degree, providing a unified framework for the analysis and construction of orthonormal bases and recurrence relations; for unitary matrices this transfers the Szegő recurrence and CMV framework to yield efficient orthogonalization procedures.

C2weakest assumption

The no-deflation assumption on the block Krylov process, which is invoked to guarantee that the generated subspace has full dimension and that the isometric isomorphism to the matrix-polynomial space holds without breakdown (abstract, paragraph on polynomial block Krylov subspaces).

C3one line summary

Block Krylov subspaces correspond isometrically to matrix polynomial spaces, allowing transfer of Szegő recurrences and CMV frameworks to orthogonalize polynomial and extended block Krylov bases for unitary matrices.

References

27 extracted · 27 resolved · 0 Pith anchors

[1] A. C. Antoulas.Approximation of large-scale dynamical systems, volume 6 ofAdvances in Design and Control. Society for Industrial and Applied Mathematics, 2005 2005

[2] E. Carson, K. Lund, M. Rozloˇ zn´ ık, and S. Thomas. Block Gram-Schmidt algorithms and their stability proper- ties.Linear Algebra Appl., 638:150–195, 2022 2022

[3] D. Damanik, M. Embree, and J. Fillman. Gap labels for zeros of the partition function of the 1D Ising model via the Schwartzman homomorphism.Indag. Math. (N.S.), 35(5):813–836, 2024 2024

[4] D. Damanik, A. Pushnitski, and B. Simon. The analytic theory of matrix orthogonal polynomials.Surv. Approx. Theory, 4:1–85, 2008 2008

[5] V. Druskin and L. Knizhnerman. Extended Krylov subspaces: approximation of the matrix square root and related functions.SIAM J. Matrix Anal. Appl., 19(3):755–771, 1998 1998

Formal links

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Receipt and verification

First computed	2026-05-20T00:03:32.737305Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

8e1506fdea0fd05744be05f1150d705fa4dfc321ab050c39c0730ef45187bd3b

Aliases

arxiv: 2605.16954 · arxiv_version: 2605.16954v1 · doi: 10.48550/arxiv.2605.16954 · pith_short_12: RYKQN7PKB7IF · pith_short_16: RYKQN7PKB7IFORF6 · pith_short_8: RYKQN7PK

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/RYKQN7PKB7IFORF6AXYRKDLQL6 \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: 8e1506fdea0fd05744be05f1150d705fa4dfc321ab050c39c0730ef45187bd3b

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "bc4e5d1d62f1fb967a202bc51b31bf557cad54b0efc3f9d10a35f98e99a54feb",
    "cross_cats_sorted": [
      "cs.NA"
    ],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.NA",
    "submitted_at": "2026-05-16T12:11:31Z",
    "title_canon_sha256": "343438720a38e7a86951387ca3193e4bc8b23d7b956832338f09cd0d3d3a8ce9"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.16954",
    "kind": "arxiv",
    "version": 1
  }
}