Element-wise Convergence Behavior of Subspace Iteration

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 $\mathrm{\Lambda }=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}({\lambda }_1,\dots ,{\lambda }_n)$ with $\mathrm{\mid }{\lambda }_1\mathrm{\mid }>\cdots >\mathrm{\mid }{\lambda }_n\mathrm{\mid }>0$, 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 $\mathrm{\mid }{Q}_k(i,j)\mathrm{\mid }$ in the subspace iteration without the Rayleigh–Ritz procedure: entries with $i\mathrm{\ne }j$ decay as $(\mathrm{\mid }{\lambda }_{\max \{i,j\}}\mathrm{\mid }\mathrm{/}\mathrm{\mid }{\lambda }_{\min \{i,j\}}\mathrm{\mid }{)}^k$, and the deviation of $\mathrm{\mid }{Q}_k(j,j)\mathrm{\mid }$ from $1$ decays as $\max \{\mathrm{\mid }{\lambda }_j\mathrm{\mid }\mathrm{/}\mathrm{\mid }{\lambda }_{j-1}\mathrm{\mid },\mathrm{\mid }{\lambda }_{j+1}\mathrm{\mid }\mathrm{/}\mathrm{\mid }{\lambda }_j\mathrm{\mid }{\}}^{2k}$, with explicit coefficients determined by the LU factorization of $X$. For the Rayleigh–Ritz variant, we obtain element-wise bounds for the normalized Ritz vectors $Z_k$. Specifically, for $\mathrm{\mid }{Z}_k(i,j)\mathrm{\mid }$, off-diagonal entries with $i\ge m+1$ decay as $(\mathrm{\mid }{\lambda }_i\mathrm{\mid }\mathrm{/}\mathrm{\mid }{\lambda }_j\mathrm{\mid }{)}^k$, off-diagonal entries with $i\le m$ decay as $(\mathrm{\mid }{\lambda }_{m+1}{\mathrm{\mid }}^2\mathrm{/}(\mathrm{\mid }{\lambda }_i\mathrm{\mid }\mathrm{\mid }{\lambda }_j\mathrm{\mid }){)}^k$, and the deviation of $\mathrm{\mid }{Z}_k(j,j)\mathrm{\mid }$ from $1$ decays as $(\mathrm{\mid }{\lambda }_{m+1}\mathrm{\mid }\mathrm{/}\mathrm{\mid }{\lambda }_j\mathrm{\mid }{)}^{2k}$. These results give an element-wise description of the convergence behavior of subspace iteration. After an orthogonal or unitary change of basis, the results apply to real symmetric or complex normal matrices.