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#### Problems

4.1 Show that the set of equations

$\Sigma \mathbit{\theta }=\mathbit{p}$ has a unique solution if Σ > 0 and infinite many if Σ is singular.

4.2 Show that the set of equations

$\Sigma \mathbit{\theta }=\mathbit{p}$ always has a solution.

4.3 Show that the shape of the isovalue contours of the mean-square error (J(θ)) surface

$J\left(\mathbit{\theta }\right)=J\left({\mathbit{\theta }}_{*}\right)+{\left(\mathbit{\theta }-{\mathbit{\theta }}_{*}\right)}^{\text{T}}\Sigma \left(\mathbit{\theta }-{\mathbit{\theta }}_{*}\right)$ are ellipses whose axes depend on the eigenstructure of Σ.

Hint

Assume that Σ has discrete eigenvalues.

4.4 Prove that if the true relation between ...

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