On symmetrizing a matrix
The paper presents an algorithm to compute a matrix symmetrizer. A symmetrizer of an arbitrary square matrix A is a symmetric matrix X satisfying the matrix equation \(XA=A^ tX\), where t indicates the transpose. It is shown that a symmetrizer reduces a nonsymmetric real eigenvalue problem into a symmetric one, which is relatively easier to solve. Moreover, it is shown that a symmetrizer can be used to obtain the zeros of a real polynomial. It is also useful in stability problems of control theory. The paper presents an algorithm to compute a symmetrizer, which requires a finite number of arithmetic operations, superimposed on a method of solving linear equations. This algorithm includes as a special case a recursive algorithm previously known for lower Hessenberg matrices. The stability of the algorithm and an example are discussed.
scientific article; zbMATH DE number 4058750