Dear Mike,

appreciated for your concern.

I have already tried to do it with (PhiTMPhi) and I had the identity matrix just as you did.

The purpose of my inquiry is to address the definition of orthogonal vectors as presented in textbooks, which states that orthogonal vectors are those whose dot product yields zero. This condition results in the identity matrix when all the eigenmodes of the matrix Phi are multiplied by themselves via dot product. However, in the context being discussed, this condition does not hold. I would like to inquire about potential scholarly references pertaining to the relationship denoted as PhiTMPhi. Alternatively, could you provide any APDL code that may be utilized to compute the modal assurance criterion (MAC) for the aforementioned modes?