There is no assumption with quadratic discriminant analysis that the groups have equal covariance matrices. As with linear discriminant analysis, an observation is classified into the group that has the smallest squared distance. However, the squared distance does not simplify into a linear function, thus the name quadratic discriminant analysis.
Unlike linear distance, quadratic distance is not symmetric. In other words, the quadratic discriminant function of group i assessed with the mean of group j is not equal to the quadratic discriminant function of group j assessed with the mean of group i. On the results, quadratic distance is called the generalized squared distance. If the determinant of the sample group covariance matrix is less than one, the generalized squared distance can be negative.
Minitab calculates the Mahalanobis distances using the individual class covariance matrices. Minitab does not calculate a quadratic discriminant function.
Use a linear analysis when you assume the covariance matrices are equal for all groups. Use a quadratic analysis when you assume the covariance matrices are not equal for all groups.