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Consider $$f(x)=\frac{1}{2}x^TQx-x^Tb$$

where $Q$ is an $n \times n$ symmetric matrix

The contours of $f$ are $n-$dimensional ellipsoids with axes in the directions of the $n$-mutually orthogonal eigenvectors of $Q$. What is the meaning of this statement ?

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    Try to work it out for the special case of $2\times 2$ matrix. – Mhenni Benghorbal Sep 24 '13 at 06:24
  • Which parts of the statement don't you understand? Do you know what an $n$-dimensional ellipsoid is? Do you know what an axis of such an ellipsoid is? Do you know what an eigenvector is? Do you know what mutually orthogonal means? – Robert Israel Sep 24 '13 at 06:26
  • I am guessing that $Q$ is positive definite. Take $Q$ to be diagonal and $b=0$ to start. – copper.hat Sep 24 '13 at 06:29
  • Take $Q=identity$ and $n=2$ and $b=[{1\atop 1}]$ $x=[{x_1\atop x_2}]$. Then $f(x)=1/2(x_1^2+x_2^2)-(x_1+x_2)$. Contour level means solution of $f(x)=C$ for some number $C$. Now in this case you get a simple circle. Next try $Q$ a diagonal matrix with diagonal entries $1,2$ to see what happens. – Maesumi Sep 24 '13 at 06:38

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