The EigenvaluesTutor(M) command presents the techniques used in finding the eigenvalues of the square matrix  by:

Creating the matrix M - lambda*Id where Id is an identity matrix with dimensions equal to that of M

Taking the determinant of M - lambda*Id

Finding the roots of the resulting characteristic polynomial

The Matrix M must be square and of dimension 4 at most.

Floating-point numbers in M are converted to rationals before computation begins.

If the symbolic expression representing an eigenvalue grows too large, then the value displayed in the Maplet application window is a floating-point approximation to it (obtained by applying evalf).  The underlying computations continue to be performed using exact arithmetic, however.

The EigenvaluesTutor(M) command returns the eigenvalues as a column Vector.