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.
Properties of unilevel block circulants
Let A={A0,A1,…,Ak-1}⊂Cd1×d2,ζ=e-2πi/k,Fl=∑m=0k-1ζlmAm,0⩽l⩽k-1, and FA=⊕l=0k-1Fl.All operations in indices are modulo k.It is well known that if d1=d2=1 then [As-r]r,s=0k-1=ΦFAΦ∗, where Φ=1k[ζlm]l,m=0k-1.However, to our knowledge it has not been emphasized that FA plays a fundamental role in connection with all the matrices [As-αr]r,s=0k-1,0⩽α⩽k-1, with d1,d2 arbitrary.We begin by adapting a theorem of Ablow and Jenner with d1=d2=1 to the case where d1 and d2 are arbitrary.We show that A=[As-αr]r,s=0k-1 if and only if A=UαFAP∗ where Uα and P are related to Φ,P is unitary, and Uα is invertible (in fact, unitary) if and only if gcd(α,k)=1, in which case we say that A is a proper circulant.We prove the following for proper circulants A=[As-αr]r,s=0k-1:(i) A†=[Br-αs]r,s=0k-1 with Bm=1k∑l=0k-1ζlmFl†,0⩽m⩽k-1.(ii) Solving Az=w reduces to solving Flul=vαl,0⩽l⩽k-1, where v0,v1,…,vk-1 depend only on w.(iii) A singular value decomposition of A can be obtained from singular value decompositions ofF0,F1,…,Fk-1. (iv) The least squares problem for A reduces to independent least squares problems for F0,F1,…,Fk-1. (v) If d1=d2=d, the eigenvalues of [As-r]r,s=0k-1 are the eigenvalues of F0,F1,…,Fk-1, and the corresponding eigenvectors of A are easily obtainable from those of F0,F1,…,Fk-1. (vi) If d1=d2=d and α>1 then the eigenvalue problem for [As-αr]r,s=0k-1 reduces to eigenvalue problems for d×d matrices related to F0,F1,…,Fk-1 in a manner depending upon α.
Noctilucent cloud observations over Greenland by a Rayleigh lidar
We report on the detection of noctilucent clouds (NLCs) by a Rayleigh lidar system located on the west coast of Greenland during operations in July and August 1994. The lidar detected NLCs in the zenith on three nights (July 28–29, July 31–August 1, and August 4). The lidar measurements are presented at temporal (1 minute) and spatial (192 meter) resolutions exceeding previously reported lidar measurements of NLCs and reveal significantly more structure in the clouds. Unique cloud features are observed for each night NLCs were detected. Because of the large backscatter ratio associated with NLCs, the volume backscatter coefficient for the cloud particles can be determined more accurately from the lidar signal than the backscatter ratio as the upper mesospheric molecular number density is negligible in the volume backscatter coefficient calculation. The volume backscatter coefficient values determined from the three nights of NLC observations would be classified by current models as moderate to extreme. The extreme backscatter from NLC measurements on August 4 tests the assumptions used in existing models.