The asymptotic behavior of the singular values of matrix powers and applications (Q1923187)

The behavior of the singular values \(s_k (A^n)\), \(k=1,2, \dots,r\), of the \(n\)th power of an \(r \times r\) matrix \(A\) is analyzed for \(n\to \infty\). One of the main results gives for each \(k\) an explicit construction of a sequence of the form \((a(k)^n n^{l(k)})^\infty_{n = 1}\) which is equivalent to the sequence \((s_k(A^n))^\infty_{n=1}\), where \(s_1(A^n)\), \(s_2(A^n), \dots, s_r(A^n)\) are the singular values of \(A^n\), and \(a(k) = a(k,A)\), \(l(k) = l(k,A)\) are independent of \(n\). Some analogous results are derived for the continuous case, where \(A^n\) is replaced by \(\exp (tA)\). In addition, this paper contains some applications to difference and differential equations with constant coefficients and the connections with similarity for the block shift.