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
Graduate students’ experiences with standardized patients as adjuncts for teaching pelvic examinations
Purpose: To explore graduate nurse practitioner students’ perceptions of their experiences when learning to perform pelvic examinations in the laboratory and performing them in subsequent clinical rotations. One group was taught by faculty with voluntary peer examination, and the other two groups were taught by standardized patients (SPs). Data sources: Surveys with open‐ and closed‐ended (responses on Likert scales) items administered twice during 3 consecutive years to students enrolled in an advanced health assessment course. Conclusions: All groups reported feeling anxious while learning pelvic exam techniques and in subsequent clinical experiences. SPs provided immediate feedback to students, decreased their feelings of anxiety, and increased their confidence in performing examinations. Students who were taught pelvic examination techniques by SPs rated their learning experiences more positively and reported a better understanding of exam techniques than students who learned to perform exams by voluntary examination of classmates. Implications for practice: Graduate nursing programs should consider locating and using SP programs for teaching pelvic examinations in advanced health assessment courses. Although more cost‐effective, voluntary peer examination was a less effective teaching method.