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Intro to Econometric Theory \linebreak
Heinz School, Carnegie Mellon University \linebreak
90-906, Spring 2005-6 \linebreak
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Homework \#1, due Friday, February 24, 2006
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\begin{enumerate}
\item Consider two matrixes: $A$, $m \times n$, and $B$, 
      $n \times p$.  Prove that their product, $AB$ can 
      be written:

      $AB = \sum_{i=1}^n A_iB^i$.

      How could we write $X'X$?

\item Prove that the commutative law of mulitplication applies
      to diagonal matrixes.  What is the inverse of a diagonal
      matrix?

\item Consider the sum $\sum_{i=1}^n(X_i-\overline{X})(Y_i-\overline{Y})$,
      where $X_i,Y_i$ are scalars.

      \begin{enumerate}
      \item Please write the sum in matrix notation, using no sums.
            (Hint: use $M_0$).
      \item Use the matrix notation and facts you know about $M_0$
            to prove:

            \begin{eqnarray}
                  \sum_{i=1}^n(X_i-\overline{X})(Y_i-\overline{Y})
                  &=&\sum_{i=1}^n(X_i-\overline{X})Y_i \nonumber\\
                  &=&\sum_{i=1}^nX_i(Y_i-\overline{Y}) \nonumber
            \end{eqnarray}

            and that
            \begin{math}
            \sum_{i=1}^n(X_i-\overline{X})(Y_i-\overline{Y})=
            \sum_{i=1}^nX_iY_i
            \end{math}

            if either the mean of $X$ or the mean of $Y$ is zero.
      \end{enumerate}
\end{enumerate}

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