where JL and JN are the reversal matrices of dimensions (L × L) and (N × N), respectively.
where C(t i ), D(t i ), E(t i ), and F(t i ) are matrices of dimensions 2 × l, 2 × l, 2 × l', and 2 × l', respectively.
with a i = ∏ j = 1 M σ i, j M, where the sequence { σ i, j } j = 1 M is the singular value of H i. U1 and U2 are unitary matrices of dimensions N1 × N1 and N2 × N2, respectively, D1 and D2 are generalized upper triangular matrices with equal diagonal elements, and † denotes the conjugate transpose.
This model has the following form: (13) where, A and B are the first and second design matrices of dimensions (k × m) and (k × n), respectively, with n ≥ k ≥ m, x is the vector of unknowns, ε is a residual vector, w is the vector of mis-closures, Q is the covariance matrix of observations and E stands for the statistical expectation.
We need to find a control (u(t)) and corresponding trajectory (x t)) that minimize the functional J=frac{1}{2} int_{0}^{T} bigl[x'(t)Q t)x t)+u'(t)C t)u(t) bigr],dt, (3) where (Q t)=Q'(t ge0), (C t)=C'(t)>0) are the matrices of dimensions (ntimes n) and (mtimes m), correspondingly.
Matrix factorization is the process where a matrix is decomposed into two matrices linked through a latent space of predefined dimension: X approx UV^{T}where (X), (U), and (V) are matrices of dimensions (n times m), (n times k), and (m times k), respectively.
