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.
Reconceptualizing Vagrancy and Reconstructing the Vagrant: A Socio-Legal Analysis of Criminal Law Reform in Canada, 1953-1972
This article explores significant reforms to the vagrancy section of the Criminal Code during the mid-to-late twentieth century. By locating the reforms within their unique social, political, and economic climates, I examine how they reconceptualized the offence of vagrancy and concomitantly reconstructed the vagrant as a social problem. The reforms played a seminal role in reducing the number of vagrancy offences, eventually leading to the demise of vagrancy in the criminal law. Yet, while the "vagrant" ceased to exist in the law, the law still continues to preserve vestiges of the vagrant in a highly gendered manner.