The title says it all. Why does positive semi definiteness implies positivity on diaginal elements.
Why does positive definiteness implies positivity on diagnal
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matrices
1 Answers
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If $A = (a_{ij})$ is a positive definite matrix then $v^T A v > 0$ for every vector $v\neq 0$. In particular, $a_{ii} = e_i^T A e_i > 0$, where $e_i= \begin{pmatrix} 0\\ \vdots\\ 0\\ 1\\ 0\\ \vdots\\ 0 \end{pmatrix} $ is the vector whose $i$-th coordinate is 1, and all other coordinates are 0.
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0$a_{ii} = e_i^T A e_i$ by the definition of matrix multiplication; e_i^T A e_i > 0 since $A$ is positive definite. – 2012-11-21