A normal matrix is a special type of matrix that has a number of synonyms. One common synonym is an "orthogonal matrix," which is a matrix that preserves a vector's length and angle. Another synonym is a "unitary matrix," which is a complex matrix whose inverse is equal to its conjugate transpose. A "self-adjoint matrix" is another synonym for a normal matrix that has the property of being equal to its own transpose. Finally, a "Hermitian matrix" is a complex self-adjoint matrix where the diagonal elements are real and the off-diagonal elements are the complex conjugates of each other. These synonyms are all related to the fundamental properties of normal matrices and provide a nuanced way to differentiate between different types of matrices and their uses in various applications.