A remark on symmetric circulant matrices
For a real symmetric nonsingular circulant \(n\times n\) matrix A the author shows that \(A^{-1}=(1/n)Q,\) where Q is an \(n\times n\) circulant matrix whose 0th row is \((q_ 0,q_ 1,...,q_{n-1})\) with \(q_ i=\sum^{n-1}_{j=0}(1/\mu_ j)(\bar w^ i)^ j,\) \(w=\exp (2\pi i/n)\), and \(\mu_ 0,\mu_ 1,...,\mu_{n-1}\) the eigenvalues of A (Theorem 1). After that the author considers the same problem for \(A=G_ 1\otimes...\otimes G_ s\), the Kronecker product of \(p_ i^{k_ i}\times p_ i^{k_ i}\) symmetric circulant matrices \(G_ i\) for \(i=1,2,...,s\), where \(n=p_ 1^{k_ 1}...p_ s^{k_ s}\) is the prime-power decomposition of n. He proves that there exists an \(n\times n\) permutation matrix T such that \(TAT^{-1}\) is symmetric (Theorem 2), and if \(Q_ i\) is nonsingular for all i's, then \(A^{-1}=T^{- 1}UD^{-1}U^*T,\) where U is the Fourier matrix of order n (i.e. \(U=(1/\sqrt{n})\cdot (w_{ij})\), \(w_{ij}=(w^ i)^ j)\), and D is the diagonal matrix consisting of the products of eigenvalues of \(G_ 1,...,G_ s\) on the main diagonal (Corollary to Theorem 2). Finally, for a nonsingular symmetric circulant block matrix B of type (m,n), i.e. an \(mn\times mn\) matrix \[ B= \left[\begin{matrix} A_ 0& A_ 1& ...& A_{m-1} \\ A_{m-1} & A_ 0& ...& A_{m-2} \\ \vdots & \vdots && \vdots \\ A_ 1& A_ 2& ...& A_ 0 \end{matrix}\right] \] where each block \(A_ i\) is a real symmetric circulant matrixwith \(A_{m-i}=A_ i\) for \(i=0,1,...,m-1\), and m,n\(\geq 2\), the author shows that \(B^{- 1}=(V\otimes U)D^{-1}(V^*\otimes U^*),\) where V is the Fourier matrix of order m, and D is a diagonal matrix. We remark that the Lemma 5 characterising symmetric circulant block matrix of type (m,n) is false. This matrix should be defined in this manner.
nonsingular symmetric circulant block matrix
scientific article; zbMATH DE number 4055793