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I was referring to this lecture http://www.stanford.edu/class/ee364a/videos/video04.html. and he gave an example of a generalized quadratic equation

f(x,y) = x'Ax + 2x'By + y'Cy

The functions is convex if the matrix

A B

B' C

is positive semidefinite and also the matrix C. I didn't get how this is derived and how the function f(x,y) is a generalized quadratic equation

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up vote 2 down vote accepted

Consider $z=\begin{pmatrix} x \\ y\end{pmatrix}$, then $f(x,y)=g(z)$ with $g(z)=z'Mz$ for $M=\begin{pmatrix} A & B \\ B' & C\end{pmatrix}$. Thus, $f$ is quadratic. And $f$ is convex if an only if $M$ is positive semidefinite (and this condition implies that $C$ is positive semidefinite as well hence there is no need to add this further condition).

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