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Inner Product Spaces
Published in Lina Oliveira, Linear Algebra, 2022
A common mistake. It is frequent that the orthogonal complement of a plane in ℝ3 (containing (0,0,0)), is confused with a (any) straight line perpendicular to that plane. Observe that, as seen in Proposition 6.9, the orthogonal complement is itself a subspace. Hence the orthogonal complement must contain (0,0,0).
Matrices and Linear Algebra
Published in William S. Levine, Control System Fundamentals, 2019
Suppose V is a vector space with an inner product, and let W be a subspace of V. The subspace W⊥ = {v: 〈v, w〉 = 0 for all w ∈ W} is called the orthogonal complement of W in V. W ∩ W⊥ = {0}, and V = W ⊕ W⊥ is an orthogonal direct sum decomposition of V: every v ∈ V can be written uniquely in the form v = w + w⊥, where w ∈ W and w⊥ ∈ W⊥. The linear function pW: V → V defined in terms of the unique decomposition, v ↦ w, is called the orthogonal projection of V onto W. The orthogonal direct sum decomposition of V may be rewritten as V = im pW ⊕ ker pW. The complementary orthogonal projection of pW is the orthogonal projection of V onto W⊥, or pW⊥; its kernel and image provide another orthogonal direct sum representation of V, whose terms correspond to the image and kernel of pW, respectively.
The elasticity complex: compact embeddings and regular decompositions
Published in Applicable Analysis, 2022
The inner product and the norm in a Hilbert space are denoted by and , respectively. We use for the orthogonal complement of a subspace of . The notations , , indicate the sum, the direct sum, the orthogonal sum of two subspaces , of , respectively. Throughout this paper, the domain of definition, the kernel, and the range of a linear operator are denoted by , , and , respectively.
A conjugate directions-type procedure for quadratic multiobjective optimization
Published in Optimization, 2022
Ellen H. Fukuda, L. M. Graña Drummond, Ariane M. Masuda
Given , span represents the subspace generated by the elements in M. For a subspace , stands for its orthogonal complement with respect to , i.e. . The Paretian cone is . For a function and , the set of minimizers of h restricted to W is denoted by . If this set has a single element, say , we write .
Control systems analysis for the Fornasini-Marchesini 2D systems model – progress after four decades
Published in International Journal of Control, 2018
Krzysztof Galkowski, Eric Rogers
A 2D system described by (2) is asymptotically stable if and only if (54) holds,there exists matrices and such that where ⊥ denotes the orthogonal complement.