Some results on certain generalized circulant matrices
An \(m \times n\) \((m\leq n)\) matrix \(A=[a_{ij}]\) is called \((h,r)\)-circulant if for each row \(i\) \((i\geq 2 )\), \(a_{ij} = a_{i-1, j-h}\), where \(j > h\) and the last \(h\) entries of the preceding row are multiplied by \(r\). It is called \(h\)-circulant and \(r\)-circulant, when \(h=1\) and \(r=1\), respectively. If \((a_{0}, a_{1},\dots,a_{n-1})\) is the first row of an \(r\)-circulant matrix \(A\) of order \(n\), then the matrix \(A\) can be written as \(A=\sum_{i=0}^{n-1}a_{i}P^{i}_{n}(r)\), where \(P_{n}(r)\) is the \(r\)-circulant matrix of order \(n\) with the first row \((0, 1, 0, \dots, 0)\). An \(n\times n\) matrix \(A\) is called \((h,r)\)-generalized circulant if it can be partitioned into \(k\) sub-matrices of order \(m\times n\) \((h,r)\)-circulant matrices, where \( 1\leq h<n\), \(k=(n,h)\), \(n=km\) and \(h=kh'\) for positive integers \(h,n\). The first set of results concerns the generalized \((h,r)\)-circulant matrices. Theorem 1 gives a characterization of this type of circulant matrices. The result is then used in Theorem 2 to show that an \(r\)-circulant matrix \(A=\sum_{i=0}^{t}a_{i}P^{ih}(r)\) is similar to a \(k\)-block diagonal matrix, \[ \text{diag} \left(\sum_{i=0}^{t}a_{i}P_{m}^{i}(r^{\theta}), \dots, \sum_{i=0}^{t}a_{i}P_{m}^{i}(r^{\theta})\right), \] where \(t=\lfloor\frac{n}{h}\rfloor\) and \(\theta=\frac{h}{k}\). The results are then used to determine the permanent and eigenvalues of the \(r\)-circulant matrix. The second set of results is related to the sparse \(r\)-circulant matrices, \(A=a_{0}I+a_{i}P_{n}^{i}(r)+a_{j}P^{j}_{n}(r)\), where \(i,j\) and \(n\) are positive integers and \(1\leq i< j\leq n\). The author gives a characterization of this type of circulant matrices under different assumptions and determines a lower bound of the permanent.
\((h,r)\)-generalized circulant matrices
scientific article; zbMATH DE number 6418185