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Given $z_t=\sum_{i=1}^t \theta_iz_{t-i}+v_t $, where $t=1,...,N$ where $N=1024$. I need to write this in matrix form (a system of linear equations) as $\mathsf{A}\mathsf{z} = \mathsf{z} - \mathsf{v}$. In this case we have that $\mathsf{z}$ and $\mathsf{v}$ are column vectors with 1024 members and the $\mathsf{A}$ matrix is a $1024 \times 1024$ matrix. I want to find $\mathsf{A}$.

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Based on your equation:

$$z_{t} = \sum_{i=1}^{t} \theta_{i} z_{t-i} + v_{t}$$

From $\mathsf{z}-\mathsf{v} = \mathsf{A}\mathsf{z}$, You have this:

$$z_{t} - v_{t} = \sum_{i=1}^{N} \mathsf{A}_{ti} z_{i}$$

Your original equation can be rewritten as:

$$z_{t} - v_{t} = \sum_{i=0}^{t-1} \theta_{t-i} z_{i} + \sum_{i=t}^{N} 0 z_{i}$$

Lets define $\Theta_{i}^{t}$ as:

$$\Theta_{i}^{t} = \begin{cases} \theta_{t-i} & i < t \\ 0 & \text{otherwise} \end{cases}$$

So:

$$z_{t}-v_{t} = \sum_{i=0}^{N} \Theta_{i}^{t} z_{i}$$

as a result: $\mathsf{A}_{ij} = \Theta_{j}^{i}$ and $z_{i}-v_{i} = \sum_{j=1}^{N} \mathsf{A}_{ij} z_{j}$

Finally:

$$(\mathsf{I}-\mathsf{A}) \mathsf{z} = \mathsf{v}$$

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