Assume I have a $n\times n$ positive semidefinite matrix $G$ of rank $p$ satisfying a set of $np - p(p-1)/2$ equations $v^T\_jGv\_j = 1$, $j = 1 \ldots np - p(p-1)/2$ for some given vectors $v\_j$. It is assumed these equations are linearly independent. Note here that the number of equations is exactly equal to the degrees of freedom in $G$. Is it then true that there are only finitely many matrices $G$ satisfying these equations? Thanks in advance.



Unique matrix satisfying a system of equations
In the case $n=3$,$p=2$, your 5 constraints for $v\_1 = (1,0,0)^T$, $v\_2 = (0,1,0)^T$, $v\_3 = (0,0,1)^T$, $v\_4 = (1,-2,0)^T$ and $v\_5 = (1,-1,1)^T$ have solution $G = \pmatrix{1 & 1 & t\cr
1 & 1 & t\cr t & t & 1\cr}$, which has rank 2 and is positive semidefinite if $-1 < t < 1$.


