Learning parameters of noise and filter coefficients from data where data and noise both have Gaussian distributions

Question

Assume $X$ and $N$ are two sets of vectors (observations) from a normal distribution, where $X$ represents clean data and $N$ represents noise; and $A$ a projection matrix of a filter. The scenario is that our clean data was corrupted by a multiplicative noise via matrix $A$ and an additive noise of $N$:

$$Y=A X + N.$$

How can we learn the projection matrix $A$ and $N$ from the training data $X,Y$? Does the Gaussian assumption of $A$, $N$ and $X$ help to have a better estimation or guide to use a specific solution?

Here is matlab code for the training data, noise and a simple projection:

    dataVariance = .10;
    noiseVariance = .05;  
    mixtureCenters=randn(13,1);
    X=randn(13, 1000)*sqrt(dataVariance ) + repmat(mixtureCenters,1,1000);

    %N and A are unknown and we want to estimate them.
    N=randn(13, 1000)*sqrt(noiseVariance ) + repmat(mixtureCenters,1,1000);
    A=2*eye(13);

    Y=A*X+N;

    for iter=1:1000
        A_hat,N_hat = training(X_hat,X,Y);
    end

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Note: if necessary, for each estimation of $A$, an error can be calculated for an estimation of $N$ using a current $A$.

For example:

for iterate=1:1000
  initiate A
  estimate N using current A (N=Y-A*X)
  calculate error of estimation (err=Y-A*X+N)
  update A

But I would prefer not to go for gradient descent approaches.