# Solved – Is the joint distribution of two linear combinations of Gaussians still a multivariate normal

Suppose I have $$textbf{X}$$ ~ $$N(textbf{0},Sigma)$$, and I'm considering two different linear combinations, $$a^* X$$ and $$b^* X$$, which we suppose are uncorrelated. I understand that linear combinations of variables in multivariate Gaussians are Gaussian, and that in a multivariate Gaussian, that uncorrelatedness implies independence.

But can I conclude from uncorrelatedness of $$a^* X$$ and $$b^* X$$ that they are independent? Do I know that $$a^* X$$ and $$b^* X$$ together form a multivariate Gaussian?

Note that uncorrelatedness of $$a^* X$$ and $$b^* X$$ does not imply a and b are orthogonal, as in 2D for a correlated Gaussian we could have a = <1,0> and $$b = $$, which are uncorrelated but not orthogonal.

Thanks.

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Do I know that a∗X and b∗X together form a multivariate Gaussian?

Yes. Form a matrix, $$M$$ as:

begin{align} M &= left[ array{a\b} right] end{align}

Since $$X$$ is Gaussian and since linear combinations of Gaussian are Gaussian, $$MX$$ is Gaussian. Since $$MX$$ is:

begin{align} MX &= left[ array{a*X\b*X} right] end{align}

we know that $$aX$$ and $$bX$$ are jointly Gaussian.

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