<s>[INST] Suppose $X, Y$ are random variables with the equal variance. Show that $X-Y$ and $X+Y$ are uncorrelated. Suppose that $X$ and $Y$ are random variables with the equal variance.
Show that $X-Y$ and $X+Y$ are uncorrelated.
I get I should use the equation  $$E[XY] = E[X]E[Y]$$ For the first part I get $$E[(X-Y)(X+Y)] = E[X^2-Y^2] = E[X^2] - E[Y^2]$$ And I don't know how to follow. Someone has any ideas? 
Thank you.
 [/INST] Suppose that
$$
\begin{pmatrix}
X\\
Y
\end{pmatrix}
$$
is a random vector with the covariance matrix
$$
\begin{pmatrix}
\sigma^2&\sigma_{X,Y}\\
\sigma_{X,Y}&\sigma^2
\end{pmatrix}.
$$
The covariance matrix of the random vector
$$
\begin{pmatrix}
1&1\\
1&-1
\end{pmatrix}
\begin{pmatrix}
X\\
Y
\end{pmatrix}
=
\begin{pmatrix}
X+Y\\
X-Y
\end{pmatrix}
$$
is given by
$$
\begin{pmatrix}
1&1\\
1&-1
\end{pmatrix}
\begin{pmatrix}
\sigma^2&\sigma_{X,Y}\\
\sigma_{X,Y}&\sigma^2
\end{pmatrix}
\begin{pmatrix}
1&1\\
1&-1
\end{pmatrix}
=
\begin{pmatrix}
2(\sigma^2+\sigma_{X,Y})&0\\
0&2(\sigma^2-\sigma_{X,Y})
\end{pmatrix}.
$$
This shows that $X+Y$ and $X-Y$ are uncorrelated and gives their variances.
</s>