2.7 Exercises

EXERCISE 2.1   Compute the determinant for a $(3\times 3)$ matrix.

EXERCISE 2.2   Suppose that $\vert\data{A}\vert = 0$. Is it possible that all eigenvalues of $\data{A}$ are positive?

EXERCISE 2.3   Suppose that all eigenvalues of some (square) matrix ${\data A}$ are different from zero. Does the inverse $\data{A}^{-1}$ of $\data{A}$ exist?

EXERCISE 2.4   Write a program that calculates the Jordan decomposition of the matrix

\begin{displaymath}\data{A} = \left(\begin{array}{ccc} 1&2&3\\ 2&1&2\\ 3&2&1
\end{array}\right) .\end{displaymath}

Check Theorem 2.1 numerically.

EXERCISE 2.5   Prove (2.23), (2.24) and (2.25).

EXERCISE 2.6   Show that a projection matrix only has eigenvalues in $\{0,1\}$.

EXERCISE 2.7   Draw some iso-distance ellipsoids for the metric $\data{A}=\Sigma^{-1}$ of Example 3.13.

EXERCISE 2.8   Find a formula for $\vert\data{A}+aa^{\top}\vert$ and for $(\data{A}+aa^{\top})^{-1}.$ (Hint: use the inverse partitioned matrix with $\data{B}=\left(\begin{array}{ll}
1 & -a^{\top}\\
a & \data{A}
\end{array}\right).$)

EXERCISE 2.9   Prove the Binomial inverse theorem for two non-singular matrices $ \data{A}( p \times p) $ and $\data{B} (p \times p)$: $(\data{A}+\data{B})^{-1}=\data{A}^{-1}-\data{A}^{-1}(\data{A}^{-1}+\data{B}^{-1})^{-1}
\data{A}^{-1}.$ (Hint: use (2.26) with ${\cal C}=\left(\begin{array}{ll}
\data{A} & I_p\\
-I_p & \data{B}^{-1}
\end{array}\right).$)