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d 11 = M − N − T + 1 − t r ( X α ′ P X α Σ ) d 12 = t r ( S ν Σ X α ′ P X α Σ ) d 13 = t r ( S λ Σ X α ′ P X α Σ ) d 21 = T − t r ( X α ′ P λ X α Σ ) d 22 = T − 2 t r ( X α ′ P λ D N D N ′ X α Σ ) + t r ( X α ′ P λ X α Σ S ν Σ ) d 23 = M − 2 t r ( S λ Σ ) + t r ( X α ′ P λ X α Σ S λ Σ ) d 31 = N − t r ( X α ′ P ν X α Σ ) d 32 = M − 2 t r ( S ν Σ ) + t r ( X α ′ P ν X α Σ S ν Σ ) d 33 = N − 2 t r ( X α ′ P ν D T D T ′ X α Σ ) + t r ( X α ′ P ν X α Σ S λ Σ )

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\begin{align*} d_{11} & = M - N - T + 1 - \mr{tr}\left(\mb{X}_\alpha ^{'}\mb{P} \mb{X}_\alpha \bSigma \right) \\ d_{12} & = \mr{tr}\left(\mb{S}_\nu \bSigma \mb{X}_\alpha ^{'}\mb{P} \mb{X}_\alpha \bSigma \right) \\ d_{13} & = \mr{tr}\left(\mb{S}_\lambda \bSigma \mb{X}_\alpha ^{'}\mb{P} \mb{X}_\alpha \bSigma \right) \\ d_{21} & = T - \mr{tr}\left(\mb{X}_\alpha ^{'}\mb{P}_\lambda \mb{X}_\alpha \bSigma \right) \\ d_{22} & = T - 2\mr{tr}\left(\mb{X}_\alpha ^{'} \mb{P}_\lambda \mb{D}_ N \mb{D}_ N^{'}\mb{X}_\alpha \bSigma \right) + \mr{tr}\left(\mb{X}_\alpha ^{'} \mb{P}_\lambda \mb{X}_\alpha \bSigma \mb{S}_\nu \bSigma \right) \\ d_{23} & = M - 2\mr{tr}\left(\mb{S}_\lambda \bSigma \right) + \mr{tr}\left(\mb{X}_\alpha ^{'} \mb{P}_\lambda \mb{X}_\alpha \bSigma \mb{S}_\lambda \bSigma \right) \\ d_{31} & = N - \mr{tr}\left(\mb{X}_\alpha ^{'}\mb{P}_\nu \mb{X}_\alpha \bSigma \right) \\ d_{32} & = M - 2\mr{tr}\left(\mb{S}_\nu \bSigma \right) + \mr{tr}\left(\mb{X}_\alpha ^{'} \mb{P}_\nu \mb{X}_\alpha \bSigma \mb{S}_\nu \bSigma \right) \\ d_{33} & = N - 2\mr{tr}\left(\mb{X}_\alpha ^{'} \mb{P}_\nu \mb{D}_ T \mb{D}_ T^{'}\mb{X}_\alpha \bSigma \right) + \mr{tr}\left(\mb{X}_\alpha ^{'} \mb{P}_\nu \mb{X}_\alpha \bSigma \mb{S}_\lambda \bSigma \right) \end{align*}
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