Quaternions are a 4-parameter representation of a rotation + scaling operator, the analog of using a complex number to represent rotation + scaling in the plane

Quaternions are a 4-parameter representation of a rotation + scaling operator, the analog of using a complex number to represent rotation + scaling in the plane. You can convert a (4 parameter) quaternion to a (9 parameter) 3x3 matrix in a similar way that you can convert a (2 parameter) complex number to a (4 parameter) 2x2 matrix. In either case you aren't using the full parameter space of the matrices. In general in programs quaternions are used for storing and manipulating rotation operators (e.g. composing them together, etc.) because it saves work and improves numerical stability, but then final application of the rotation to a gigantic list of vectors is done using a matrix, because matrix-vector multiplication is very efficient (only need 9 multiplications and 6 additions to multiply a 3x3 matrix by a 3-element vector).