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.
Characterization and properties of matrices with k-involutory symmetries II
We say that a matrix R ∈ C is k-involutory if its minimal polynomial is x − 1 for some k ≥ 2, so R = R and the eigenvalues of R are 1, ζ, ζ,. . . , ζ, where ζ = e. Let α, μ ∈ {0, 1, . . . , k − 1}. If R ∈ C, A ∈ C, S ∈ C and R and S are k-involutory, we say that A is (R, S, α,μ)-symmetric if RAS = ζA. We show that an (R, S, α, μ)-symmetric matrix A can be usefully represented in terms of matrices F` ∈ C cα`+μ×d` , 0 ≤ ` ≤ k − 1, where c` and d` are respectively the dimensions of the ζeigenspaces of R and S. This continues a theme initiated in an earlier paper with the same title, in which we assumed that α = 1. We say that a k-involution is equidimensional with width d if all of its eigenspaces have dimension d. We show that if R and S are equidimensional k-involutions with widths d1 and d2 respectively, then (R, S, α, μ)-symmetric matrices are closely related to generalized αcirculants [ζAs−αr] k−1 r,s=0, where A0, A1, . . . , Ak−1 ∈ C d1×d2 . For this case our results are new even if α = 1. We also give an explicit formula for the Moore-Penrose inverse of a unilevel block circulant [As−αr] k−1 r,s=0 for any α ∈ {0, 1, . . . , k − 1}, generalizing a result previously obtained for the case where gcd(α, k) = 1. MSC: 15A09; 15A15; 15A18; 15A99
Characterization and Properties of Matrices with Generalized Symmetry or Skew Symmetry
Let R∈Cn×n be a nontrivial involution; i.e., R=R−1≠±I. We say that A∈Cn×n is R-symmetric (R-skew symmetric) if RAR=A (RAR=−A). 
 
There are positive integers r and s with r+s=n and matrices P∈Cn×r and Q∈Cn×s such that P∗P=Ir, Q∗Q=Is, RP=P, and RQ=−Q. We give an explicit representation of an arbitrary R-symmetric matrix A in terms of P and Q, and show that solving Az=w and the eigenvalue problem for A reduce to the corresponding problems for matrices APP∈Cr×r and AQQ∈Cs×s. We also express A−1 in terms of APP−1 and AQQ−1. Under the additional assumption that R∗=R, we show that Moore–Penrose inversion and singular value decomposition of A reduce to the corresponding problems for APP and AQQ. We give similar results for R-skew symmetric matrices. These results are known for the case where R=J=(δi,n−j+1)i,j=1n; however, our proofs are simpler even in this case. 
 
We say that A∈Cn×n is R-conjugate if RAR=A where R∈Rn×n and R=R−1≠±I. In this case R(A) is R-symmetric and I(A) is R-skew symmetric, so our results provide explicit representations for R-conjugate matrices in terms of P and Q, which are now in Rn×r and Rn×s respectively. We show that solving Az=w, inverting A, and the eigenvalue problem for A reduce to the corresponding problems for a related matrix S∈Rn×n. If RT=R this is also true for Moore–Penrose inversion and singular value decomposition of A.