As we know, there are only $\frac{n(n+1)}{2}$ variables in a symmetric $n$-dimensional semi-definite matrix. Is it possible to produce a $n$-dimensional semi-definite matrix whose trace is $1$ using only $\frac{n(n+1)}{2}-1$ variables, $\frac{n(n+1)}{2}$ from symmetry and one from the trace constraint?
If so, what is the best way?
Till now, I can produce a $n$-dimensional semi-definite matrix $\rho$ whose trace is $1$ in the way
t = ({
{a, b, c, d},
{e, f, g, h},
{i, j, k, l},
{m, n, o, p}
});
ρ = t.t\[Transpose]/Total[Flatten[t]^2];
but I need $n^2$ variables and sometimes Total[Flatten[t]^2]
may be zero because of semi-definiteness. Any help or suggestions will be appreciated.