Linear Systems, Eigenvalues, and Projection
The development of Model Order Reduction techniques for various problems was triggered by the success of subspace projection methods for the solution of large linear systems and for the solution of matrix eigenvalue problems.
The most well-known approaches in the subspace projection arena are based on the construction of Krylov subspaces. These subspaces were proposed in 1931 by Krylov for the explicit construction of the characteristic polynomial of a matrix, so that the eigenvalues could be computed as the roots of that polynomial. This initial technique proved to be unpractical for matrices of order larger than, say, 6 or 7. It failed because of the poor quality of the standard basis vectors for the Krylov subspace. An orthogonal basis for this subspace appeared to be an essential factor, as well as the way in which this orthogonal basis is generated. These were breakthroughs initiated by Lanczos  and Arnoldi , both in the early 1950’s. In this chapter we will first discuss briefly some standard techniques for solving linear systems and for matrix eigenvalue problems.We will mention some relevant properties, but we refer the reader for background and more references to the standard text by Golub and van Loan .
We will then focus our attention on subspace techniques and highlight ideas that are relevant and can be carried over to Model Order Reduction approaches for other sorts of problems.
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