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into a matrix Φ allows the matrices M and K to be diagonalized. Speciﬁcally, the eigenvectors are mass-normalized such that: ΦT MΦ = I and ΦT KΦ = Ω2 (3) The matrix Φ can be intuitively described as a matrix which trans-forms between object space and mode space: each column of Φ contains the shape of a normal mode, while ΦT f ...