Preliminaries from Matrix Theory
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In this chapter we review the matrix theory used in this book. The purpose of the first section is to introduce the notation and to call to memory results derived in any core curriculum with a basic matrix theory course. The other three sections are small excursions to obtain theorems often not included in such a course. First the fact that every matrix is similar to a matrix in pseudo-diagonal normal form is established. It is the pivotal theorem in the convergence proof for annealing chains which is given in chapter 3. Another fact used in that proof is Gershgorin’s theorem. Its statement and proof is at the end of the third section. That section is mainly devoted to the convergence of matrix expressions and since norms are useful concepts in that context they are introduced there. That section completes the preparation for the next chapter. The definite integral obtained in the fourth section of this chapter is not used before chapter 6. It is included in this chapter because its derivation heavily depends on matrix theory, and quadratic forms in particular.
KeywordsQuadratic Form Matrix Theory Triangular Matrix Matrix Norm Kronecker Product
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