Linear recursive sequences and powers of matrices (Q2746563)
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scientific article; zbMATH DE number 1656224
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | Linear recursive sequences and powers of matrices |
scientific article; zbMATH DE number 1656224 |
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20 January 2003
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linear recurrence sequence
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power of matrix
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characteristic polynomial
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generating function
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Stirling numbers
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Cayley-Hamilton theorem
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Linear recursive sequences and powers of matrices (English)
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The author gives basic properties for linear recursive sequences \(\{u_n(a_1, a_2,\ldots, a_n)\}\) of the form \(u_n+a_1u_{n-1}+\cdots+a_mu_{n-m}\) \((n=0, \pm 1, \pm 2,\ldots)\), \(u_{1-m}=\cdots=u_{-1}=0\), \(u_0=1\). The author also gives a formula for powers of matrices as NEWLINE\[NEWLINE A^n=\sum_{r=0}^{m-1}\left(\sum_{s=r}^{m-1}a_{s-r}u_{n-s}\right)A^r, NEWLINE\]NEWLINE where \(A\) is an \(m\times m\) matrix with characteristic polynomial \(x^m+a_1x^{m-1}+\cdots+a_m\).
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