## 3 Mode Pairing Using the Energy-Based Modal Assurance Criterion (EMAC)
### Complex Modes
Therefore, in the case of complex modes, the modal strain energy can be calculated based on the second orthogonality condition, and the total strain energy associated with mode \(j\) (MSE\({}_{j}\)) is equal to \(1/2\times b_{j}\). The matrix \(\boldsymbol{\Psi}\) has a dimension of \(2z\times 2n^{\prime}\), where \(z\) is the number of degrees of freedom of the numerical model and \(n^{\prime}\) is the total number of vibration modes. This matrix can be rearranged by separating the degrees of freedom of the numerical model into \(n\) clusters, resulting in:  
\[\boldsymbol{\Psi}= \begin{bmatrix}\boldsymbol{\Phi}_{11}&\boldsymbol{\Phi}_{12 }&\cdots&\boldsymbol{\Phi}_{1n^{\prime}}&\boldsymbol{\Phi}_{11}^{*}& \boldsymbol{\Phi}_{12}^{*}&\cdots&\boldsymbol{\Phi}_{1n^{\prime}}^{*}\\ \boldsymbol{\Phi}_{21}&\boldsymbol{\Phi}_{22}&\cdots&\boldsymbol{\Phi}_{2n^ {\prime}}&\boldsymbol{\Phi}_{21}^{*}&\boldsymbol{\Phi}_{22}^{*}&\cdots& \boldsymbol{\Phi}_{2n^{\prime}}^{*}\\ \vdots&\vdots&\ddots&\vdots&\vdots&\vdots&\ddots&\vdots\\ \boldsymbol{\Phi}_{n1}&\boldsymbol{\Phi}_{n2}&\cdots&\boldsymbol{\Phi}_{nn^ {\prime}}&\boldsymbol{\Phi}_{n1}^{*}&\boldsymbol{\Phi}_{n2}^{*}&\cdots& \boldsymbol{\Phi}_{nn^{\prime}}^{*}\\ \boldsymbol{\Phi}_{11}\lambda_{1}&\boldsymbol{\Phi}_{12}\lambda_{2}&\cdots& \boldsymbol{\Phi}_{1n^{\prime}}\lambda_{n^{\prime}}&\boldsymbol{\Phi}_{11}^{* }\lambda_{1}^{*}&\boldsymbol{\Phi}_{12}^{*}\lambda_{2}^{*}&\cdots&\boldsymbol {\Phi}_{1n^{\prime}}^{*}\lambda_{n^{\prime}}^{*}\\ \boldsymbol{\Phi}_{21}\lambda_{1}&\boldsymbol{\Phi}_{22}\lambda_{2}&\cdots& \boldsymbol{\Phi}_{2n^{\prime}}\lambda_{n^{\prime}}&\boldsymbol{\Phi}_{21}^{* }\lambda_{1}^{*}&\boldsymbol{\Phi}_{22}^{*}\lambda_{2}^{*}&\cdots&\boldsymbol {\Phi}_{2n^{\prime}}^{*}\lambda_{n^{\prime}}^{*}\\ \vdots&\vdots&\ddots&\vdots&\vdots&\ddots&\vdots\\ \boldsymbol{\Phi}_{n1}\lambda_{1}&\boldsymbol{\Phi}_{n2}\lambda_{2}&\cdots& \boldsymbol{\Phi}_{nn^{\prime}}\lambda_{n^{\prime}}&\boldsymbol{\Phi}_{n1}^{* }\lambda_{1}^{*}&\boldsymbol{\Phi}_{n2}^{*}\lambda_{2}^{*}&\cdots&\boldsymbol {\Phi}_{nn^{\prime}}^{*}\lambda_{n^{\prime}}^{*}\end{bmatrix}  \tag{23}\]  
The vector that contains the modal information of the numerical mode \(j\) can be rearranged by separating the degrees of freedom of the \(n\) clusters. It has the following format:  
\[\boldsymbol{\Psi}_{j}^{T}= \begin{bmatrix}\boldsymbol{\Phi}_{1j}^{T} \boldsymbol{\Phi}_{2j}^{T}\cdots\boldsymbol{\Phi}_{nj}^{T}\big{(}\boldsymbol {\Phi}_{1j}\lambda_{j}\big{)}^{T}\big{(}\boldsymbol{\Phi}_{2j}\lambda_{j} \big{)}^{T}\cdots\big{(}\boldsymbol{\Phi}_{nj}\lambda_{j}\big{)}^{T}\end{bmatrix}  ^{T} \tag{24}\]In turn, the matrix \(\mathbf{Q}\), with a dimension of \(2z\times 2z\), is also divided into submatrices (\(\mathbf{K}_{kl}\) and \(\mathbf{M}_{kl}\)) that relate the degrees of freedom of clusters \(k\) and \(l\), that is:  
\[\mathbf{Q}= \begin{bmatrix}\mathbf{K}_{11}&\mathbf{K}_{12}&\dots&\mathbf{K}_{1n }&\mathbf{0}&\mathbf{0}&\dots&\mathbf{0}\\ \mathbf{K}_{21}&\mathbf{K}_{22}&\dots&\mathbf{K}_{2n}&\mathbf{0}&\mathbf{0}& \dots&\mathbf{0}\\ \vdots&\vdots&\ddots&\vdots&\vdots&\vdots&\ddots&\vdots\\ \mathbf{K}_{n1}&\mathbf{K}_{n2}&\dots&\mathbf{K}_{nn}&\mathbf{0}&\mathbf{0}& \dots&\mathbf{0}\\ \mathbf{0}&\mathbf{0}&\dots&\mathbf{0}&-\mathbf{M}_{11}&-\mathbf{M}_{12}& \dots&-\mathbf{M}_{1n}\\ \mathbf{0}&\mathbf{0}&\dots&\mathbf{0}&-\mathbf{M}_{21}&-\mathbf{M}_{12}& \dots&-\mathbf{M}_{2n}\\ \vdots&\vdots&\ddots&\vdots&\vdots&\vdots&\ddots&\vdots\\ \mathbf{0}&\mathbf{0}&\dots&\mathbf{0}&-\mathbf{M}_{n1}&-\mathbf{M}_{n2}& \dots&-\mathbf{M}_{nn}\end{bmatrix}  \tag{25}\]  
in which \(k\) and \(l\) can assume values equal to \(1,\;2,\;\dots,\;n\), where \(n\) is the total number of clusters. Based on this submatrix division of \(\mathbf{Q}\), it is possible to form the matrices \(\mathbf{Q}_{kl}\), which relate the degrees of freedom from clusters \(k\) and \(l\), that is:  
\[\mathbf{Q}_{kl}= \begin{bmatrix}\mathbf{K}_{kl}&0\\ 0&-\mathbf{M}_{kl}\end{bmatrix}