On tridiagonal matrices unitarily equivalent to normal matrices

In this article the unitary equivalence transformation of normal matrices to tridiagonal form is studied. It is well-known that any matrix is unitarily equivalent to a tridiagonal matrix. In case of a normal matrix the resulting tridiagonal inherits a strong relation between its superand subdiagonal elements. The corresponding elements of the superand subdiagonal will have the same absolute value. In this article some basic facts about a unitary equivalence transformation of an arbitrary matrix to tridiagonal form are firstly studied. Both an iterative manner based on Krylov sequences as a direct manner via Householder transformations are deduced. This equivalence transformation is then reconsidered for the normal case and equality of the absolute value between the superand subdiagonals is proved. Self-adjointness of the resulting tridiagonal matrix with regard to a specific scalar product will be proved. Flexibility in the reduction will then be exploited and properties when applying the reduction on symmetric, skew-symmetric, Hermitian, skew-Hermitian and unitary matrices and their relations with, e.g., complex-symmetric and pseudo-symmetric matrices are presented. It will be shown that the reduction can then be used to compute the singular value decomposition of normal matrices making use of the Takagi factorization. Finally some extra properties of the reduction as well as an efficient method for computing a unitary complex symmetric decomposition of a normal matrix are given.