Matrix decomposition techniques: use and limitation in modal analysis

Many activities in modal analysis research and applications are classified under the general area of the inverse problem. Examples are modal parameter identification, parameter identification, finite element model updating and damage detection, among others. These type of problems usually result in a system of linear equations that are characterized by one or both of the following: (a) ill conditioned system of equations, (b) Underdetemined system of equations. Matrix decomposition or factorization techniques can be extremely helpful in dealing with or detecting singularities and numerical ill conditioning. Among these methods are: (a) QR and QL algorithms, (b) LU factorization, (c) Chokesky decomposition, (d) Singular Value Decomposition. This paper is to introduce the theoretical grounds of these mathematical tools. Application of these techniques in modal analysis is illustrated and limitations and usefulness of solutions are emphasized.