arXiv:2609.21540v3 Announce Type: replace Abstract: This paper studies the element-wise convergence behavior of subspace iteration. All results are established for both \(\mathbb{F}=\mathbb{R}\) and \(\mathbb{F}=\mathbb{C}\). For a diagonal matrix \(\Lambda=\mathrm{diag}(\lambda_1,\ldots,\lambda_n)\) with \(|\lambda_1|>\cdots>|\lambda_n|>0\), 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 \(X\in\mathbb{F}^{n\times m}\), we first derive exact asymptotic expressions for \(|Q_k(i,j)|\) in the subspace iteration without the Rayleigh--Ritz procedure: entries with \(i\neq j\) decay as \(|\lambda_{\max\{i,j\}}/\lambda_{\min\{i,j\}}|^k\), and the deviation of \(|Q_k(j,j)|\) from \(1\) decays as \(\max\{|\lambda_j/\lambda_{j-1}|,|\lambda_{j+1}/\lambda_j|\}^{2k}\), 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 asymptotic formulas for the normalized Ritz vectors \(Z_k\). Specifically, for \(|Z_k(i,j)|\), off-diagonal entries with \(i\geq m+1\) decay as \(|\lambda_i/\lambda_j|^k\), off-diagonal entries with \(i\leq m\) decay as \(|\lambda_{m+1}^2/(\lambda_i\lambda_j)|^k\), and the deviation of \(|Z_k(j,j)|\) from \(1\) decays as \(|\lambda_{m+1}/\lambda_j|^{2k}\). 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.