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$x_1, x_2 > 0$

$f(x_1,x_2) = \frac{1}{x_1^3x_2}$

Is this function convex?

I think it is because Hessian matrix is $$\begin{pmatrix} \frac{12}{x_1^5x_2} & \frac{3}{x_1^4x_2^2} \\ \frac{3}{x_1^4x_2^2} & \frac{2}{x_1^3x_2^3}\end{pmatrix}$$

How do i show that the matrix is positive? multiply it by vector a^T and a isn't working..

Thanks!

can i just say its positive cause every element is positive? enter image description here

Adar
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  • Do you know the connection between positive definiteness and the determinants of the principal minors? – Martin R Jun 08 '20 at 11:43
  • https://en.wikipedia.org/wiki/Definiteness_of_a_matrix#Characterizations – Martin R Jun 08 '20 at 11:46
  • My fault, that applies only to nondecreasing positive convex functions. – Martin R Jun 08 '20 at 11:47
  • so the matrix is positive because the 1x1 matrix is positive? – Adar Jun 08 '20 at 11:48
  • Not just the $1\times 1$ matrix, you need to look at both the upper left $1\times 1$ and the full matrix. See https://en.wikipedia.org/wiki/Sylvester%27s_criterion. – Minus One-Twelfth Jun 08 '20 at 11:56
  • By the way, just because a matrix has all non-negative entries, it doesn't mean that the matrix is positive semi-definite (consider for instance the matrix $A = \begin{pmatrix}1 & 2 \ 2 & 1\end{pmatrix}$). The 'non-negative matrix' definition you showed is something different to the definition of 'positive semi-definite'. – Minus One-Twelfth Jun 08 '20 at 11:59
  • can u please show me the calculation that needs to be done in order to show that the matrix is positive definite? (or non negative).. i really don't understand what it is write in the wiki page.. please – Adar Jun 08 '20 at 12:01
  • You need to show that the upper left $1\times 1$ submatrix and the upper left $2\times 2$ submatrix (which is just the whole matrix) both have positive determinant. Are you able to calculate the determinants of these and show that they're positive? Note also that the determinant of a $1\times 1$ matrix is just the element itself. – Minus One-Twelfth Jun 08 '20 at 12:05
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    now i udnerstand what to do.. i did it thanks.. and no, I've no idea what the guy told me in the last post.. I've no idea.. – Adar Jun 08 '20 at 13:16

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