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I have a problem. From wikipedia http://en.wikipedia.org/wiki/Positive-definite_matrix any function can be written as $$z^TMz$$ where z is a column vector and M is a symmetric real matrix. However this quadratic function is strictly convex only when M is symmetric positive definite. Why ?, I thought any quadatic function should be convex ? doesn't $$z^TMz$$ >0 shows only that the range of this function is greater than zero? 1. Why isn't any symmetric matrix M(which represents a quadratic function) convex ?

  1. Why is it only the case when $$z^TMz$$ denotes convex ?

Thanks

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Suppose $M=\pmatrix{1&0\cr0&-1\cr}$. Then $M$ is symmetric and real, and $z^tMz=x^2-y^2$. Is $x^2-y^2$ convex? What's your definition of convex? – Gerry Myerson Oct 10 '12 at 1:17
For function to be convex I mean: en.wikipedia.org/wiki/Convex_function $$f(tx_1+(1-t)x_2) <tf(x_1) +(1-t)f(x_2) $$ for any 0 < t < 1, x1 != x2. – Jing Oct 10 '12 at 1:24
OK, so, does $f=x^2-y^2$ fit that definition? What if you pick points with $f(x_1)=f(x_2)=0$, say? – Gerry Myerson Oct 10 '12 at 1:30
hmm... I am confused, can you give me some sketch of mathematical proof? f(x) is convex iff M(f(x)) is SGD ? thankx – Jing Oct 10 '12 at 1:42
First, have you found an example to show that for the $M$ I gave, the function is not convex? Second, what is SGD? – Gerry Myerson Oct 10 '12 at 1:59

1 Answer

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For any twice differentiable function, it is convex if and only if, the Hessian matrix is positive definite. You can find it from any standard textbook on convex optimization. Now here the function at hand is $z^TMz$ which is clearly twice differentiable (by virtue of being quadratic). Now the Hessian of this function is $M$ (please verify yourself, it helped me a lot to memorize it). So $M$ should be positive definite for that quadratic function to be convex.

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Thank you ! that's very clear. – Jing Oct 10 '12 at 7:04

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