Under ideal conditions, the matrix is diagonalized and the remaining factors can be merged with.
For the first case, as the right-hand side matrix is diagonalized, the solution of (2)is as (1).
Equation (10) results from the fact that addition and removal of CP turns channel matrix into a circulant matrix, and the resulting circulant matrix is diagonalized by DFT processing [24].
2. Tensor reconstruction was performed based on DTI, and tensor matrix was diagonalized to obtain eigenvalues λ1, λ2 and λ3, as well as corresponding eigenvectors, then the FA value of each voxel was calculated according to the following formula.
After this process, the diffusion tensor elements were estimated by solving the Stejskal and Tanner equation [23], [24], and then the reconstructed tensor matrix was diagonalized to obtain three eigenvalues (λ1, λ2, λ3) and eigenvectors.
The columns were mean-centered, matrix multiplication was applied, and the product matrix was diagonalized yielding the eigenvectors and eigenvalues.
After this process, the diffusion tensor elements were estimated by solving the Stejskal and Tanner equation [ 42, 43], and then the reconstructed tensor matrix was diagonalized to obtain three eigenvalues (λ1, λ2, and λ3) and eigenvectors.
Following the fact that circulant matrices can be diagonalized by DFT matrices, say F, reconsider the term E[u k u k T K k u k u k T ], remembering that a process linearly filtered by a unitary filter F for very long filters preserves its properties (see Appendix 2 for proof) at the output of the filter.
We notice that any circulant matrix can be diagonalized as where is the diagonal matrix whose diagonal elements are the DFT of the first column of.
Moreover, circulant matrices can be diagonalized using a discrete Fourier transform (DFT) as mathbb{G} = mathbb{W}^{H} mathbb{D} mathbb{W} (20).
