Download Matrix Computations and Semiseparable Matrices - Linear by Raf Vandebril PDF

By Raf Vandebril

In contemporary years numerous new periods of matrices were came upon and their constitution exploited to layout quickly and exact algorithms. during this new reference paintings, Raf Vandebril, Marc Van Barel, and Nicola Mastronardi current the 1st complete evaluation of the mathematical and numerical homes of the family's most up-to-date member: semiseparable matrices.

The textual content is split into 3 components. the 1st presents a few ancient historical past and introduces suggestions and definitions relating dependent rank matrices. the second one bargains a few conventional tools for fixing structures of equations concerning the fundamental subclasses of those matrices. The 3rd part discusses dependent rank matrices in a broader context, provides algorithms for fixing higher-order established rank matrices, and examines hybrid versions reminiscent of block quasiseparable matrices. An available case research essentially demonstrates the overall subject of every new notion mentioned. a number of the workouts featured are applied in Matlab and will be downloaded from the internet for additional exploration.

(Spring 2009)

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28 Consider the S +D : ⎡ 1 ⎢ ⎢ S +D =⎢ 2 ⎣ 3 1 is not correct, however, as the following example following 4 × 4 semiseparable plus diagonal matrix 3 9 2 3 6 3 1 1 3 ⎤ ⎡ ⎥ ⎢ ⎥ ⎢ ⎥+⎣ 1 ⎦ 4 ⎤ 1 2 ⎥ ⎥. ⎦ 1 1−3 1 The limit gives us the following matrix: ⎡ 2 ⎢ 2 lim (S + D ) = ⎢ ⎣ 3 →∞ 0 3 3 3 0 9 6 1 1 ⎤ 0 0 ⎥ ⎥, 1 ⎦ 5 which is neither a tridiagonal, nor a semiseparable plus diagonal matrix. The matrix however is quasiseparable. Moreover, we remark once more that the resulting matrix has some kind of symmetry in the rank structure.

1 a b a b 0 0 0 bd d ⎦ + ⎣ 0 c − bd 0 ⎦⎠ = ⎣ b c d ⎦ lim ⎝⎣ b →∞ 1 d e 0 0 0 d e The matrices on the left are of semiseparable plus diagonal form, whereas the limit is a tridiagonal matrix, which does not belong to the class of semiseparable plus diagonal matrices. (d) Before proving the main theorem of this subsection, namely that S sym = Qsym , we show which quasiseparable matrices are not of semiseparable plus diagonal form. 2. 19. A symmetric quasiseparable matrix A ∈ Qsym is not representable as a semiseparable plus diagonal matrix if and only if there exists an i ∈ {2, .

In [260], Strang proves a related result and he comments on different ways to prove the nullity theorem. The following corollary is a straightforward consequence of the nullity theorem. 36 (Corollary 3 in [114]). Suppose A ∈ Rn×n is a nonsingular matrix, and α, β are nonempty subsets of N with |α| < n and |β| < n. Then rank A−1 (α; β) = rank (A(N \β; N \α)) + |α| + |β| − n. Proof. By permuting the rows and columns of the matrix A, we can always move the submatrix A(N \β; N \α) into the upper left part A11 .

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