Take the 2-minute tour ×
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.

This question comes from projective geometry. A degenerate conic $C$ is defined as $$C=lm^T+ml^T,$$

where $l$ and $m$ are different lines. It can be easily shown, that all points on $l$ and m lie on the $C$. Because, for example, if $x\in l$, then by definition $l^Tx=0$ and plugging it into conic equation makes it true.

Question: Find the rank of $C$.

(We can limit ourself to 3-dimensional projective space.)

P.S. I'm reading a book, where it is guessed and checked, but I would like to have a proof without guessing. I do not provide the guess, since it can distract you, but if you really need it just leave a note.

share|improve this question

1 Answer 1

The rank of $lm^T$ is one. The same goes for $ml^T$. In most cases, the rank of the symmetric matrix $C$ as you define it will be 2. This corresponds to a conic degenerating into two distinct lines. If the lines $l$ and $m$ should coincide, though, the rank of $C$ will be 1.

If you need a proof, you can show this assumption for specific cases without loss of generality. Of you can have a look at the corresponding dual conics and how that relates to adjoint matrices.

share|improve this answer

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.