Element-wise Convergence Behavior of Subspace Iteration
Abstract
This paper studies the element-wise convergence behavior of subspace iteration. All results are established for both 𝔽=ℝ and 𝔽=ℂ. For a diagonal matrix Λ=diag(λ1,…,λn) with |λ1|>⋯>|λn|>0, we analyze the modulus of each entry of the iterative matrix sequence generated by subspace iteration, both without and with the Rayleigh--Ritz procedure. Under mild assumptions on the initial matrix X∈𝔽n× m, we first derive exact asymptotic expressions for |Qk(i,j)| in the subspace iteration without the Rayleigh--Ritz procedure: entries with i≠ j decay as |λmax{i,j}/λmin{i,j}|k, and the deviation of |Qk(j,j)| from 1 decays as max{|λj/λj−1|,|λj+1/λj|}2k, with explicit coefficients determined by the LU factorization of X. Similar conclusions could also be established for upper triangular matrices. For the Rayleigh--Ritz variant, we obtain element-wise bounds for the normalized Ritz vectors Zk. Specifically, for |Zk(i,j)|, off-diagonal entries with i≥ m+1 decay as (|λi|/|λj|)k, off-diagonal entries with i≤ m decay as (|λm+1|2/(|λi||λj|))k, and the deviation of |Zk(j,j)| from 1 decays as (|λm+1|/|λj|)2k. These results give an element-wise description of the convergence behavior of subspace iteration. After an orthogonal or unitary change of basis, the results could be further generalized.
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BibTeX
@misc{zhao2026elementwise,
title = {{Element-wise Convergence Behavior of Subspace Iteration}},
author = {Mingyang Zhao},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.21540},
eprint = {2609.21540},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.21540},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Element-wise Convergence Behavior of Subspace Iteration
AU - Mingyang Zhao
PY - 2026
DA - 2026/10/24
PB - arXiv
DO - 10.48550/arxiv.2609.21540
UR - https://arxiv.org/abs/2609.21540
AB - This paper studies the element-wise convergence behavior of subspace iteration. All results are established for both 𝔽=ℝ and 𝔽=ℂ. For a diagonal matrix Λ=diag(λ_1,…,λ_n) with |λ_1|>⋯>|λ_n|>0, we analyze the modulus of each entry of the iterative matrix sequence generated by subspace iteration, both without and with the Rayleigh--Ritz procedure. Under mild assumptions on the initial matrix X∈𝔽^n× m, we first derive exact asymptotic expressions for |Q_k(i,j)| in the subspace iteration without the Rayleigh--Ritz procedure: entries with i≠ j decay as |λ_max{i,j}/λ_min{i,j}|^k, and the deviation of |Q_k(j,j)| from 1 decays as max{|λ_j/λ_j−1|,|λ_j+1/λ_j|}^2k, with explicit coefficients determined by the LU factorization of X. Similar conclusions could also be established for upper triangular matrices. For the Rayleigh--Ritz variant, we obtain element-wise bounds for the normalized Ritz vectors Z_k. Specifically, for |Z_k(i,j)|, off-diagonal entries with i≥ m+1 decay as (|λ_i|/|λ_j|)^k, off-diagonal entries with i≤ m decay as (|λ_m+1|^2/(|λ_i||λ_j|))^k, and the deviation of |Z_k(j,j)| from 1 decays as (|λ_m+1|/|λ_j|)^2k. These results give an element-wise description of the convergence behavior of subspace iteration. After an orthogonal or unitary change of basis, the results could be further generalized.
ER -