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Applied and industrial mathematics in Italy: Proc. of the by Primicerio M., et al. (eds.)

By Primicerio M., et al. (eds.)

Commercial arithmetic is evolving into an enormous department of arithmetic. Mathematicians, in Italy particularly, have gotten more and more conscious of this new pattern and are engaged in bridging the distance among hugely really expert mathematical study and the rising call for for innovation from undefined. during this appreciate, the contributions during this quantity offer either R&D employees in with a normal view of current abilities, and lecturers with state of the art functions of arithmetic to real-world difficulties, that can even be included in complex classes.

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The function · satisfies the following conditions: ⎧ x ∈ R, ( x = 0 ⇔ x = 0), ⎨ x ≥0 x+y ≤ x + y x, y ∈ R, ⎩ cx = |c| x x ∈ R, c ∈ R. Some popular vector norms over Rn are the following ones: x 1 = (|x1 | + |x2 | + · · · + |xn |) , x 2 = |x1 |2 + |x2 |2 + · · · + |xn |2 x ∞ 1/2 , = max |xi |. 1≤i≤n These norms are equivalent as the following inequalities illustrate: x ∞ ≤ x 1 ≤ √ n x 2 ≤n x ∞. Moreover, all norms over Rn are equivalent. A set of vectors x( ) is said to converge to a vector x, if lim x( ) − x = 0.

Finally, the class of quasiseparables is defined as follows. 24. A matrix S is called a quasiseparable matrix if all the subblocks, taken out of the strictly lower triangular part of the matrix (respectively, the strictly upper triangular part) are of rank 1. Thus, a matrix A is quasiseparable, if the following relations are satisfied (for i = 1, . . , n − 1): rank (S(i + 1 : n, 1 : i)) ≤ 1, rank (S(1 : i, i + 1 : n)) ≤ 1. 25 We know that, for example, general tridiagonal and semiseparable matrices are also quasiseparable, but we can also have matrices of the following form: ⎤ ⎤ ⎡ ⎡ 1 3 3 1 1 1 ⎣ 1 1 1 ⎦ and ⎣ 4 2 3 ⎦ .

1 The nullity theorem In this section we will prove the nullity theorem in two different ways. Although this theorem is not so widely spread, it can easily be used to derive several interesting results about structured rank matrices and their inverses. It was formulated for the first time by Gustafson [167] for matrices over principal ideal domains. In [114], Fiedler and Markham translated this abstract formulation to matrices over a field. Barrett and Feinsilver formulated theorems close to the nullity theorem in [13, 15].

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