Inverse problems for unilevel block α‐circulants
We consider the following inverse problems for the class Cα of unilevel block α‐circulants C=[Cs−αr]r,s=0k−1 , where k > 1, C0, C1, …, Ck−1∈Cd1×d2 , α ∈ {1,2, … ,k −1}, gcd(α,k) = 1, and ∥ ⋅ ∥ is the Frobenius norm.
Procrustes problems and associated approximation problems for matrices with k-involutory symmetries
Abstract We say that a matrix R ∈ C n × n is k-involutary if its minimal polynomial is x k - 1 for some k ⩾ 2 , so R k - 1 = R - 1 and the eigenvalues of R are 1 , ζ , ζ 2 , … , ζ k - 1 , where ζ = e 2 π i / k . Let α , μ ∈ { 0 , 1 , … , k - 1 } . If R ∈ C m × m , A ∈ C m × n , S ∈ C n × n and R and S are k-involutory, we say that A is ( R , S , μ ) -symmetric if RAS - 1 = ζ μ A , and A is ( R , S , α , μ ) -symmetric if RAS - α = ζ μ A . Let L be the class of m × n ( R , S , μ ) -symmetric matrices or the class of m × n ( R , S , α , μ ) -symmetric matrices. Given X ∈ C n × t and B ∈ C m × t , we characterize the matrices A in L that minimize ‖ AX - B ‖ (Frobenius norm), and, given an arbitrary W ∈ C m × n , we find the unique matrix A ∈ L that minimizes both ‖ AX - B ‖ and ‖ A - W ‖ . We also obtain necessary and sufficient conditions for existence of A ∈ L such that AX = B , and, assuming that the conditions are satisfied, characterize the set of all such A.
Least-square solutions for inverse problems of centrosymmetric matrices☆
Let CSRnxn={A=(aij)ϵRnxn|aij=an+1−j, i,j=1,2…, n} In this paper, we mainly discuss solving the following problems. PROBLEM Io Given X, B ϵ Rnxm Find A ϵ CSRn×n such that AX = B. PROBLEM I. Given X, B ϵ Rn×m. Find A ϵ CSRn×n such that ∥AX−B∥=min.PROBLEM II. Given A ϵ Rn×n. Find A∗ ϵ SE such that where ∥ · ∥ is the Frobenius norm, and SE is the solution set of Problem I or I0. 
 
The expressions of the general solution of Problems I and 10 have been derived. The sufficient and necessary conditions for the solvability of Problem I0 have been provided. The numerical algorithm to find the optimal approximate solution and some numerical experiments have been given.