Tell me more ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.
up vote 0 down vote favorite

The title says it all. Why does positive semi definiteness implies positivity on diaginal elements.

share|improve this question

1 Answer

up vote 3 down vote

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.

share|improve this answer
Could you please elaborate more on $a_{ii} = e_i^T A e_i > 0$. I did not understand it. – Aman Deep Gautam Nov 15 '12 at 23:21
I added the definition of $e_i$ to my answer. Use the definition of matrix multiplication to check that $a_{ii} = e_i^T A e_i$. – Yury Nov 15 '12 at 23:27
I do not get it, this what my question says. Actually it is just mathematical way of expressing my question. What I would like to know is the proof for this. – Aman Deep Gautam Nov 16 '12 at 14:51
Could you please elaborate more on your answer. – Aman Deep Gautam Nov 20 '12 at 16:35
Explain what you don't understand. – Yury Nov 20 '12 at 17:56
show 2 more comments

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.