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Consider the first multivariate normal random variable $x \sim \mathcal{N} (x |0, \Sigma_x )$, where $x \in R^d$, $\Sigma_x \in R^{d \times d}$. We also are given another multivariate normal random variable $y \sim \mathcal{N} (y |0, \Sigma_y ) $ , $ y \in R^d$.

We would like to find a matrix $A \in R^{d \times d} $ such that $Ax$ will have similar distribution with $y$ when we already know $\Sigma_x; \Sigma_y$

We know that $Ax \sim (Ax |0, A \Sigma_x A^T ) $ so we need to find $A$ such that

$$ A \Sigma_x A^T =\Sigma_y \ (1) $$

My question here how to find $A$ based on Eq. (1)

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You could use the eigen-decomposition https://en.wikipedia.org/wiki/Eigendecomposition_of_a_matrix

$$\Sigma_x=V_x D_x V_x^T$$ $$\Sigma_y=V_y D_y V_y^T$$ where $V_x,V_y$ are matrices with orthornormal eigen-vectors and $D_x$, $D_y$ diagonal matrices of eigenvalues.

Then it is easy to write the coordinate transformation that will convert $\Sigma_x$ to $\Sigma_y$ It will be $$A=V_y \hat{D}_y\hat{D}_x^{-1}V_x^T$$ where $\hat{D}_x$,$\hat{D}_y$ are the square-roots of $D_x,D_y$, i.e. diagonal matrices formed from $D_x$,$D_y$ where elements on the diagonal are square-roots of elements from $D_x,D_y$.

Importantly this conversion is not unique. You could also use the Cholesky decomposition in the same way.

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