\[ Tr(A) = Tr(A^{T}) \]

\[ Tr(A+B) = Tr(A)+Tr(B) \]

\[ Tr(\alpha A) = \alpha Tr(A) \]

\[ Tr(AB) = Tr(BA) \] \[ Tr(BAB^{-1}) = Tr(A) \]

\[ Tr(XX^TYY^T) = Tr(Y^TXX^TY) \]

\[ X n\times p : X_{i,j} \] \[ Y n \times q : Y_{i,k} \]

\[ P q \times p : P_{k,j} : (Y^TX) \]

\[ P_{k,j} = (Y'X)_{k,j} = \sum_{r=1}^n X_{r,j}.Y_{r,k} \]

\[ R q \times n : R_{k,i} : (Y'XX') \] \[ R_{k,i} = (Y'XX')_{k,i} = \sum_{s=1}^p P_{k,s}.X_{i,s} = \sum_{s=1}^p (\sum_{r=1}^n X_{r,s}.Y_{r,k}).X_{i,s} = \sum_{s=1}^p \sum_{r=1}^n X_{r,s}.X_{i,s}.Y_{r,k} \]

\[ V q \times q : V_{k,k'} : (Y'XX'Y) \]

\[ V_{k,k'} = (Y'XX'Y)_{k,k'} = \sum_{t=1}^n R_{k,t}.Y_{t,k'} = \sum_{t=1}^n \sum_{s=1}^p \sum_{r=1}^n X_{r,s}.X_{t,s}.Y_{r,k}.Y_{t,k'} \]

\[ A n \times n : A_{i,i'} : (XX^T) \]

\[ A_{i,i'} = (XX^T)_{i,i'} = \sum_{s=1}^p X_{i,s}.X_{i',s} \]

\[ B n \times q : B_{i,k} : (XX^TY) \]

\[ B_{i,k} = (XX^TY)_{i,k} = \sum_{r=1}^n A_{i,r}.Y_{r,k} = \sum_{r=1}^n \sum_{s=1}^p X_{i,s}.X_{r,s}.Y_{r,k} \]

\[ C n \times n : C_{i,i'} : (XX^TYY^T) \]

\[ C_{i,i'} = (XX^TYY^T)_{i,i'} = \sum_{u=1}^q B_{i,u}.Y_{i',u} = \sum_{u=1}^q \sum_{r=1}^n \sum_{s=1}^p X_{i,s}.X_{r,s}.Y_{r,u}.Y_{i',u} \]

\[ Tr(C) = \sum_{\alpha=1}^n C_{\alpha,\alpha} = \sum_{\alpha=1}^n \sum_{u=1}^q \sum_{r=1}^n \sum_{s=1}^p X_{\alpha,s}.X_{r,s}.Y_{r,u}.Y_{\alpha,u} \]

\[ Tr(V) = \sum_{\beta=1}^q V_{\beta,\beta} = \sum_{\beta=1}^q \sum_{t=1}^n \sum_{s=1}^p \sum_{r=1}^n X_{r,s}.X_{t,s}.Y_{r,\beta}.Y_{t,\beta} \]

\[ Tr(C) = \sum_{\alpha=1}^n C_{\alpha,\alpha} = \sum_{\alpha=1}^n \sum_{r=1}^n \sum_{s=1}^p \sum_{u=1}^q X_{\alpha,s}.X_{r,s}.Y_{r,u}.Y_{\alpha,u} \]

\[ Tr(V) = \sum_{\beta=1}^q V_{\beta,\beta} = \sum_{t=1}^n \sum_{r=1}^n \sum_{s=1}^p \sum_{\beta=1}^q X_{r,s}.X_{t,s}.Y_{r,\beta}.Y_{t,\beta} \]

\[ Tr(C) = \sum_{t=1}^n C_{t,t} = \sum_{t=1}^n \sum_{r=1}^n \sum_{s=1}^p \sum_{u=1}^q X_{t,s}.X_{r,s}.Y_{r,u}.Y_{t,u} \] \[ Tr(V) = \sum_{u=1}^q V_{u,u} = \sum_{t=1}^n \sum_{r=1}^n \sum_{s=1}^p \sum_{u=1}^q X_{r,s}.X_{t,s}.Y_{r,u}.Y_{t,u} \]

\[ W^2 = (XX^TYY^T) \] \[ Tr(W^2) = Tr(WW) \]

Si cela peut aider, en posant que la d