Mapping between LSQ for several samples and one single sample

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I have a sysid problem of the type:

$Y = AX$,

where X is a matrix whose columns are different input vectors, and Y is the output. Therefore, A is a general matrix that maps all the input vectors(samples I have) to its output.

Image that I input one input vector at a time (columns of X), then:

$y_i = A_i x_i$

Is there a way to map A to $A_i$? Note that the matrices have the same dimension. How to assess the structure of A to find the $A_i$s?

Tks