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By F. E. Relton

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This transformation is called a similarity transformation. 4 A special type of transformation is the congruence transformation. 3. A matrix A ∈ Rn×n is said to be congruent with a matrix B ∈ Rn×n if there exists a nonsingular matrix C ∈ Rn×n such that: C T AC = B. This transformation is called a congruence transformation. 5 In this chapter we will explore all these three transformations as a preprocessing step for computing the eigenvalues and/or singular values. The clue is to perform any of these transformations, such that the obtained matrix B is of easy structure, which means that one can more efficiently compute its eigenvalues/singular values.

3. The representations 13 fusion, are also shown. Based on these graphical schemes the reduction algorithm is reinterpreted. In Chapter 8, the graphical representation is used for reconstructing the implicit QR-algorithm. , the multishift algorithm would be hard to read. Notes and references In the literature much attention is being paid to the class of generator representable semiseparable matrices. Especially as they can be considered as the inverses of irreducible tridiagonal matrices. ☞ S. O.

The Symmetric Eigenvalue Problem, volume 20 of Classics in Applied Mathematics. SIAM, Philadelphia, Pennsylvania, USA, 1998. ☞ G. W. Stewart. Matrix Algorithms, Volume II: Eigensystems. SIAM, Philadelphia, Pennsylvania, USA, 2001. ☞ L. N. Trefethen and D. Bau. Numerical Linear Algebra. SIAM, Philadelphia, Pennsylvania, USA, 1997. The original articles in which the reduction to semiseparable and semiseparable plus diagonal form were described are the following ones (see also [155]): ☞ M. Van Barel, R.

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