Consider matrix $A$ with $(j,k)$′th entry $A_{j,k}=\sin(\omega_j t_k+\phi_j),\,\forall j,k\in\{1,2,...,n\}$, where $\omega_j,t_k,\phi_j\in \mathbf R$.

1. For $t_k=k$, what is the condition on $\omega_j$, aside from the trivial one $\omega_j=q_j\pi$ for some $q_j\in\mathbf Z$, for A to be singular? What can we say about the eigenvalues and eigenvectors of A?

2. The same question as in 1) but for general $t_k\in\mathbf R$.

Thoughts: For 1), the complex variable version of the problem, i.e. $A_{j,k}=e^{i\omega_j k}$, is much simpler. $A$ is a Vandermonde matrix and is singular iff $\exists j,k, \omega_j-\omega_k=2q\pi$ for some $q\in\mathbf Z$. However, I am not clear whether and how I can convert between the complex variable version and the above real variable one. I am not aware of and can not find any results regarding the eigenvalues and eigenvectors of Vandermonde matrix.

Problem 2) seems even harder.

Consider matrix $A$ with $(j,k)$′th entry $A_{j,k}=\sin(\omega_j t_k+\phi_j),\,\forall j,k\in\{1,2,...,n\}$, where $\omega_j,t_k,\phi_j\in \mathbf R$.

1. For $t_k=k$, what is the condition on $\omega_j$, aside from the trivial one $\omega_j=q_j\pi$ for some $q_j\in\mathbf Z$, for A to be singular? What can we say about the eigenvalues and eigenvectors of A? Particularly, I am interested in whether it is true that, when $q_j\pm q_k \neq q\pi, \forall j,k\in\{1,2,...,n\}$ and $\forall q\in\mathbf Z$, $A$ is nonsingular.

2. The same question as in 1) but for general $t_k\in\mathbf R$.

Thoughts: For 1), the complex variable version of the problem, i.e. $A_{j,k}=e^{i\omega_j k}$, is much simpler. $A$ is a Vandermonde matrix and is singular iff $\exists j,k, \omega_j-\omega_k=2q\pi$ for some $q\in\mathbf Z$. However, I am not clear whether and how I can convert between the complex variable version and the above real variable one. I am not aware of and can not find any results regarding the eigenvalues and eigenvectors of Vandermonde matrix.

Problem 2) seems even harder.