Let [itex]e_i [/itex] be a unit vector with one 1 in the [itex]i[/itex]-th element. Is the following expression has a recursive presentation?(adsbygoogle = window.adsbygoogle || []).push({});

$$y_N = \sum_{k=0}^N {\frac{{{X^k} e_i}}{\|{{{X^k} e_i}\|}_2}} $$

where [itex]X[/itex] is a [itex]n \times n[/itex] square matrix, and [itex]{\| \cdot \|}_2[/itex] is a vector norm defined as [itex]{\|z\|}_2 = \sqrt{|z_1|^2+|z_2|^2+...+|z_n|^2}[/itex].

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EDIT:I know that if [itex]y_N = \sum_{k=0}^N {{X^k} e_i} [/itex], it is easy to obtain the following recursive formula:

$$y_{k+1} = X y_{k} + e_i, \quad (k=0,1,2,...) \textrm{ with } \ \ \ y_0=e_i$$

However, after we add a normalized factor, is there a similar recursive expression? Thanks.

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# Converting an explicit series to a recursive form

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