Index Of The Matrix 1999

Let ( A \in \mathbbC^n \times n ). The index of (A), denoted (\textind(A)), is the smallest nonnegative integer (k) such that

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The answer lies in the .

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: Let ( A \in \mathbbC^n \times n )