Inverse eigenvalue problems associated with spring-mass systems

AbstractIt is well known that given λ1 ⩽ … ⩽ λn and μ1 ⩽ … ⩽ μn − 1, there exists a unique n × n Jacobi matrix T such that σ(T) = {λi} and σ(T1) = {μi} (notation: Tj denotes T with row j and column j removed) if and only if λ1 < μ1 < λ2 < … < μn − 1 < λn. It was recently noticed by Gladwell that if instead of deleting the first row and column from T, we delete the jth column, then the condition stated above is sufficient for the existence of T, but if j ≠ 1 and j ≠ n, then T is not uniquely determined by σ(T) and σ(Tj) unless the spectral data satisfy some additional conditions. We give a related theorem where instead of prescribing σ(T) and σ(Tj), we prescribe σ(T) and σ(T + E) with E a certain rank one matrix. Interest in the construction of Jacobi matrices from spectral data is motivated partly by discrete spring-mass systems. Given the principal frequencies of vibration of such a system and the frequencies of a modification of the same system, then constructing a Jacobi matrix from those spectral data may lead to determining the masses and spring constants of the system. We provide general tools for converting Jacobi matrix results into spring-mass system results, illustrate these techniques with examples, among them are new spring-mass reconstruction results which follow from our Jacobi matrix theorem