Modal matrix rows

Question

A typical numerical matrix diagonalization routine (such as eigh() in numpy) returns a list of matrix eigenvalues and corresponding normalized eigenvectors in the form of a unitary "modal matrix" $U$ (whose columns they are) which diagonalizes the original matrix. Is there a simple interpretation of modal matrix rows? These rows can be thought of as columns of $U^T$, but $U^T$ is also a unitary matrix diagonalizing the original matrix. Are the rows of $U$ also transposed eigenvectors of the original matrix? If so, then what is then the eigenvalue corresponding to $n$th column?