Hello expensive customer to our community We will proffer you an answer to this query linear algebra – if $vec(W_{k-1}) = x_{k-1}otimes A^Tv_{k-1}$ Then $vec(W_k) = x_k otimes A^Tv_k$ ,and the respond will breathe typical by way of documented data sources, We welcome you and proffer you fresh questions and solutions, Many customer are questioning concerning the respond to this query.

linear algebra – if $vec(W_{k-1}) = x_{k-1}otimes A^Tv_{k-1}$ Then $vec(W_k) = x_k otimes A^Tv_k$

This query could be very unostentatious, however notation weighty. Bear with me.

We have a ceaseless matrix $A in mathbb R^{n occasions d}$ the place $d geq n$ and $rank(A) = n$.

We too have a ceaseless vector $b in mathbb R^{n occasions 1}$ and a scalar $alpha in mathbb R^+$.

We have two sequences, one is a sequence of matrices, the opposite is a sequence of vectors:

${W_k}_{ok=0}^infty subset mathbb R^{d occasions d}, {x_k}_{ok=0}^{infty} subset mathbb R^{d occasions 1}$. Where $W_0, x_0$ got to us. The relaxation of the weather of the sequences are outlined recursively:

$W_k = W_{k-1} – alpha A^T(AW_{k-1}x_{k-1}-b)x_{k-1}^T$

$x_k = x_{k-1} – alpha W^TA^T(AW_{k-1}x_{k-1}-b)$

We now want to show (or disprove) that if $vec(W_{k-1}) = x_{k-1}otimes A^Tv_{k-1}$ for some $n occasions 1$ vector $v_{t-1}$ then $vec(W_k) = x_k otimes A^Tv_k$ for some $n occasions 1$ vector $v_t$. Several numerical experiments display that is undoubted, however I’ve but to provide you with a convincing dispute.

we are going to proffer you the answer to linear algebra – if $vec(W_{k-1}) = x_{k-1}otimes A^Tv_{k-1}$ Then $vec(W_k) = x_k otimes A^Tv_k$ query through our community which brings all of the solutions from a number of dependable sources.

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