I am looking for a procedure to find solution(s) for a square matrix equation

$H^T H = S$

where $H = H^\dagger$ is a hermitian ($n\times n$) matrix and $S$ is a given symmetric complex matrix. Due to hermiticity of $H$, $S$ should satisfy $n$ conditions $(\operatorname{im}(\operatorname{tr}(S^i))=0,\quad i = 1,...,n)$.

I am interested in simple solutions for small $n=3$ matrices. For $n=2$, this can be solved by an explicit parametrization of $H$ which leads to a quadratic equation giving four solutions. Since the problem is similar to taking a square root of a matrix, presumably there are $2^n$ solutions for this problem, too.

note: This question is perhaps similar to this one. Here, the equation is simpler but applies to complex, not real matrices.

I am looking for a procedure to find solution(s) for a square matrix equation

$H^T H = S$

where $H = H^\dagger$ is a hermitian ($n\times n$) matrix and $S$ is a given symmetric complex matrix. Due to hermiticity of $H$, $S$ should satisfy $n$ conditions $(\operatorname{im}(\operatorname{tr}(S^i))=0,\quad i = 1,...,n)$.

I am interested in simple solutions for small $n=3$ matrices. For $n=2$, this can be solved by an explicit parametrization of $H$ which leads to a quadratic equation giving four solutions. Since the problem is similar to taking a square root of a matrix, presumably there are $2^n$ solutions for this problem, too.

note: This question is perhaps similar to this one. Here, the equation is simpler but applies to complex, not real matrices.

I am looking for a procedure to find solution(s) for a square matrix equation

$H^T H = S$

where $H = H^\dagger$ is a hermitian ($n\times n$) matrix and $S$ is a given symmetric complex matrix. Due to hermiticity of $H$, $S$ should satisfy $n$ conditions (im(tr($S^i$))=0, i = 1,...,n).

I am interested in simple solutions for small $n=3$ matrices. For $n=2$, this can be solved by an explicit parametrization of $H$ which leads to a quadratic equation giving four solutions. Since the problem is similar to taking a square root of a matrix, presumably there are $2^n$ solutions for this problem, too.

note: This question is perhaps similar to this one. Here, the equation is simpler but applies to complex, not real matrices.

I am looking for a procedure to find solution(s) for a square matrix equation

$H^T H = S$

where $H = H^\dagger$ is a hermitian ($n\times n$) matrix and $S$ is a given symmetric complex matrix. Due to hermiticity of $H$, $S$ should satisfy $n$ conditions $(\operatorname{im}(\operatorname{tr}(S^i))=0,\quad i = 1,...,n)$.

I am interested in simple solutions for small $n=3$ matrices. For $n=2$, this can be solved by an explicit parametrization of $H$ which leads to a quadratic equation giving four solutions. Since the problem is similar to taking a square root of a matrix, presumably there are $2^n$ solutions for this problem, too.

note: This question is perhaps similar to this one. Here, the equation is simpler but applies to complex, not real matrices.