Index Of The Matrix 1999 [ 500+ GENUINE ]

Consider the nilpotent Jordan block (J_1999(0)):

[ J = \beginpmatrix 0 & 1 & 0 & \cdots & 0 \ 0 & 0 & 1 & \cdots & 0 \ \vdots & \vdots & \ddots & \ddots & \vdots \ 0 & 0 & \cdots & 0 & 1 \ 0 & 0 & \cdots & 0 & 0 \endpmatrix_1999 \times 1999. ] index of the matrix 1999

Computational experiment (simulated):
Using double-precision arithmetic, computing (J^k) for (k>50) without reorthogonalization leads to catastrophic loss of rank information. A 1999-era algorithm would compute the numerical nullspace via SVD of (J), then restrict (J) to that subspace, iterating until the restricted matrix is numerically nonsingular. For (J_1999(0)), this requires 1999 iterations in exact arithmetic but would terminate earlier due to roundoff. Consider the nilpotent Jordan block (J_1999(0)): [ J

For an integer matrix A (m×n) of rank r, compute its Smith normal form: U A V = diag(d1, d2, …, dr, 0, …, 0), with d1 | d2 | … | dr positive integers (the invariant factors). Then: Procedure to compute index via Smith normal form:

Procedure to compute index via Smith normal form:

For a singular (M = I - P) where (P) is a stochastic matrix with absorbing states, (\textind(M)) equals the maximum length of a transient path. A chain with 1999 transient states arranged in a line (each feeding only the next) yields index 1999.