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linear algebra – Recurrence relation in matrices

I’ve the next recursive sequence:

$Z_k = Z_{k-1} – AA^TZ_{k-1}xx^T$ the place $Z_k in mathbb R^{n occasions d}, A in mathbb R^{n occasions d}, d > n, rank(A) = n, x in mathbb R^{d occasions 1}$

$A$ is a ceaseless matrix, $x$ is a ceaseless vector. Theoretically, this sequence of matrices $Z_k$ is fully clear by $Z_0$ the preliminary component of the sequence.

If I offer you $Z_0$ you’ll be able to discover $Z_k$ for any $okay$.

Suppose I wish to discover $Z_{100}$. Is it workable to seek out an expression for $Z_k$ as a duty of $Z_0$ in order that I do not even have to seek out $Z_1, Z_2, …, Z_{99}$?

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