On tridiagonal matrices unitarily equivalent to normal matrices
Two matrices \(A\) and \(B\) of the same dimensions are equivalent if nonsingular matrices \(T\) and \(S\) exist such that \(A=S^{-1}BT\). Unitarily equivalent means that both \(S\) and \(T\) are unitary. In this paper, the unitary equivalence transformation of a matrix to tridiagonal form is studied. The main results are related to applying the tridiagonalization procedure to normal matrices. The resulting tridiagonal matrix yields interesting properties related to its super- and subdiagonal elements. It is shown that these super- and subdiagonal elements have the same absolute value. A Householder tridiagonalization scheme as well as an iterative method and its relation to Krylov subspaces are presented. Several possibilities for reducing matrices are discussed and are illustrated by well-known classes of normal matrices. Specific results are obtained when the algorithm is applied to symmetric, skew-symmetric, Hermitian, skew-Hermitian and unitary matrices. Self-adjointness of the resulting tridiagonal matrix with regard to a specific scalar product is also proved. Finally, it is shown that the reduction can be used to compute the singular value decomposition of normal matrices making use of Takagi factorization. Some extra properties of the tridiagonalization procedure are presented together with the efficient method for computing a unitary complex symmetric decomposition of a normal matrix.
scientific article; zbMATH DE number 5706968
scientific article