Discussion
Let \(\underline{X} \sim N_{n}(\mu, \sigma^{2}I)\) and A be a \(n \times n\) matrix. Then \(\underline{Y} = AX\) are uncorrelated normally distributed if and only if which of the following conditions hold ?
Answer: C
The key's answer is that A is an orthogonal matrix.
\(Y = AX\) is normal with covariance \(\sigma^{2}AA'\). Its components are uncorrelated when \(AA'\) is diagonal. An orthogonal A gives \(AA' = I\), so \(Y \sim N_{n}(A\mu, \sigma^{2}I)\): the components are uncorrelated (indeed independent) with the same variance, which is the property the question has in mind.
A general non-singular A gives \(AA'\) with non-zero off-diagonal entries. An idempotent matrix of rank below \(n\) makes \(\sigma^{2}AA'\) singular and correlated in general, and a matrix of all 1s makes every component equal to \(\sum X_{i}\), perfectly correlated.
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