# Specific vectors in the null space of a complex matrix

I'm dealing here with the following problem: I have a complex matrix $F$, having size $N \times M$, with $N<<M$. I can easily compute a basis of the null space of $F$, i.e. the space of vectors $x$ such that $Fx=0_v$, where $0_v$ denotes the $N \times 1$ column vector of all zeros.

However, I'm interested to specific solutions of the $Fx=0_v$ problem, i.e. to the vector(s) $x$ such that $\left|x_i\right|=1$ for $i=1,...,M$. I'm seeking vectors having constant amplitude.

Do you have any insights? Thanks

• Are you sure about $|x_i| = 1$ for every $i=1,...,M$? Not every matrix admit such a $x$. – user251257 Jul 3 '15 at 9:42
• The amplitude should be constant, i.e. can be also different from one, while real and imaginary part can be of course different... – Garbt Jul 3 '15 at 9:51
• the matrix $(1, 0)$ does not admit such a $x$. Real or complex. – user251257 Jul 3 '15 at 9:52
• Sorry, I didn't understand your comment... What do you mean for the matrix (1,0)? – Garbt Jul 3 '15 at 9:54
• the matrix has only 1 row and the components are $1$ and $0$. The null space is $\{(0, x_2) | x_2 \in \mathbb C\}$. So there is no $x$ with the properties you want. If you meant $|x_1| = ... = |x_M|$ by the constant amplitude, then the zero vector is a trivial solution and for most the cases also the only solution. – user251257 Jul 3 '15 at 10:02