# System of bilinear equations [closed]

Given constants $n$ and $\ell$, where $\ell < n$, suppose we have the following set of equations in column vectors $a_1, a_2, \dots, a_n \in {\mathbb{C}^{\ell}}$

$$\begin{array}{rl} \langle a_1, a_n \rangle &= 0\\ \langle a_1, a_{n-1} \rangle + \langle a_2, a_n \rangle &= 0\\ \langle a_1, a_{n-2} \rangle + \langle a_2, a_{n-1} \rangle + \langle a_3, a_{n}\rangle &= 0\\ &\vdots\\ \langle a_1, a_2 \rangle + \langle a_2, a_3 \rangle + \cdots + \langle a_{n-1}, a_n \rangle &= 0\end{array}$$

where $\langle a, b \rangle := a^H b$. How can one find $a_1, a_2, \dots, a_n$ that satisfy these equations?

## closed as off-topic by Loïc Teyssier, Gerry Myerson, Nik Weaver, Stefan Waldmann, coudyDec 5 '17 at 12:39

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question does not appear to be about research level mathematics within the scope defined in the help center." – Loïc Teyssier, Gerry Myerson, Nik Weaver, Stefan Waldmann, coudy
If this question can be reworded to fit the rules in the help center, please edit the question.

• Is this a question of math research (which is the kind of question this MO website is for)? – Gerry Myerson Dec 5 '17 at 12:15
• Simulposted to m.se, with no notice to either site, math.stackexchange.com/questions/2552089/… Please don't do that. – Gerry Myerson Dec 5 '17 at 12:44
• Simultaneously posting without a cross-reference is, indeed, a misdemeanor (though, perhaps, the first offense should be forgivable). As to the question itself, it is stated not too badly except that some normalization is necessary (otherwise "Put all $a_k=0$" is an answer) and I don't see why it is trivial off hand. How about you, @GerryMyerson? (or any other one who voted to close). . – fedja Dec 5 '17 at 15:12