I know the proof of the theorem that states: the effect of applying a linear (invertible) function to a normal variable is still normal. However, I don't know how to prove that we can derive any normal distribution from the linear mapping of the standard normal distribution. Can you tell me in what form I can prove this? I decided to find a matrix A and vector b for creating a linear transformation on a standard normal distribution, and then try to show how to choose A and b to create any normal distribution with any mean and variance matrix.
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$\begingroup$ This is purely a result in linear algebra. Useful keywords to search for answers include "SVD", "QR", "Gram-Schmidt," and even "matrix square root." $\endgroup$– whuber ♦Commented Oct 17 at 13:25
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Remember the definition of normal distribution on a finite-dimensional space $(\mathbf V, \langle \cdot, \cdot\rangle): $ $\mathbf X$ follows normal distribution if for each $\mathbf u \in\mathbf V, ~\langle \mathbf u, \mathbf X\rangle $ has a normal distribution on $\mathrm{ l\!R}.$
Result: Let $\bf X$ has a normal distribution on $(\mathbf V, \langle \cdot, \cdot\rangle);$ let $\mathbf A\in\mathscr L(\mathbf V,\mathbf W),$ where $(\mathbf W, [\cdot, \cdot])$ is a finite-dimensional vector space. Then for $\mathbf w_0\in\mathbf W, ~\mathbf{AX}+\mathbf w_0$ has a normal distribution on $\mathbf W. $
Proof: Definition dictates to show that for each $\mathbf w\in\mathbf W, ~[\mathbf w, \mathbf{AX}+\mathbf w_0]$ has a normal distribution on $\mathrm{ l\!R}.$ For that, note $[\mathbf w, \mathbf{AX}+\mathbf w_0]= [\mathbf w, \mathbf{AX}]+ [\mathbf w, \mathbf w_0]=\langle \mathbf{ A^\top w},\mathbf X\rangle + \textrm{constant},$ where the former term is normal, by assumption.
$\blacksquare$
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Reference:
Multivariate Statistics: A Vector Space Approach, Morris L. Eaton, IMS, $2007, $ sec. $3.1.$
