Generalized Vandermonde tensors
The authors define an \(m\)-th-order \(n\)-dimensional real tensor \(\mathcal{A}\) to be a type-1 generalized Vandermonde tensor (GV\(_{\text{1}}\)) if there exists a vector \(v=(v_{1},\dots,v_{n})^{T}\) such that the \((i_{1},\dots,i_{m} )\)-th entry \(A_{i_{1}\dots i_{m}}\) of \(\mathcal{A}\) equals \(v_{i}^{s}\) where \(i=i_{1}\) and \(s=i_{2}+\dots+i_{m}-m+1\). They define \(\mathcal{A}\) to be a type-2 generalized Vandermonde tensor (GV\(_{\text{2}}\)) if there exists an \((m-1)\)-th-order \(n\)-dimensional tensor \(\mathcal{B}\) such that \(A_{i_{1}\dots i_{m}} =B_{i_{1}\dots i_{m-1}}^{i_{m}-1}\). For any complex vector \(x=(x_{1},\dots ,x_{n})\), \(x^{m}\) denotes the \(m\)-th-order \(n\)-dimensional tensor whose \((i_{1} ,\dots,i_{m})\)-th entry equals \(x_{i_{1}}...x_{i_{m}}\). A complex number \(\lambda\) is called an eigenvalue of \(\mathcal{A}\) if there exists a complex vector \(x=(x_{1},\dots,x_{n})\neq0\) such that the \(1\)-st-order tensor \(\mathcal{A}x^{m-1}\) equals \(\lambda(x_{1}^{m-1},\dots,x_{n}^{m-1})\). Various elementary properties of these generalized Vandermonde tensors are proved. The main theorem of the paper gives a description of the spectrum of a GV\(_{\text{1}}\)-matrix in terms of the spectrum of \(V\), the (ordinary) Vandermonde \(n\times n\) matrix whose \((i,j)\)th entry is \(v_{i}^{j-1}\). (Theorem 3). If the entries of \(v\) are distinct and nonzero (so \(\det V\neq0\)) and \(\lambda_{1},\dots,\lambda_{n}\) are the eigenvalues of \(V\), then the eigenvalues of \(\mathcal{A}\) are \(\lambda_{i}^{m-1}\) (\(i=1,\dots,n\)), each repeated \(m-1\) times.
scientific article; zbMATH DE number 6714851
scientific article