It is known that a a matrix is diagonalisable (by the similarity equivalence relation) if and only if there exists a basis of eigenvectors. A typical course in linear algebra then gives two additional equivalent statements: the geometric multiplicities and algebraic multiplicities coincide; the minimal polynomial has no repeated roots.
It is also known that a matrix is unitarily (orthogonally) diagonalisable if and only if there exists an unitarily orthonormal (orthonormal) basis if and only if the matrix is normal (normal with real entries).
What is the analogous statement for matrix congruence and matrix Hermitian congruence? Can we provide a necessary and sufficient condition for matrices to be congruently diagonalisable or Hermitely diagonalisable, i.e. what are the diagonalisable bilinear/sesquilinear forms?