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Matrices and Linear Algebra
Published in William S. Levine, The Control Handbook: Control System Fundamentals, 2017
A matrix formulation is also useful in studying orthogonal projections in the inner product space ℝn. A matrix A ∈ ℝn×n is called a projection matrix when A2 = A, which corresponds to A being the matrix representation of a projection f on an n-dimensional real vector space with respect to some basis. The matrix representation of the complementary projection is I − A. In other words, Ax is a projection defined on (the coordinate space) ℝn. For the subspace W of ℝn having an orthonormal basis {w1, …, wk}, an orthogonal projection onto W is given by WWT = A, where W ∈ ℝn×k is the matrix whose columns are the orthonormal basis vectors. Notice that this orthogonal projection matrix is symmetric. In fact, the choice of orthonormal basis does not matter; uniqueness of the orthogonal projection onto any subspace can be shown directly from the defining equations A2 = A and AT = A.
A common solution of f-fixed point and variational inequality problems in Banach spaces
Published in Optimization, 2023
Habtu Zegeye, Oganeditse A. Boikanyo
If E is a Hilbert space and , then the Bregman projection reduces to the metric projection of x onto C.If C is a smooth Banach space and , then the Bregman projection reduces to the generalized projection which is defined by where which is called the Lyapunov function, and J is the normalized duality mapping.
Interior approximate controllability of second-order semilinear control systems
Published in International Journal of Control, 2022
The operator A in (5) is the generator of semigroup which is given by where is a complete family of orthogonal projection in the Hilbert space given by and where matrix is given as also Moreover, and the eigenvalues of are and where .
Convergence results for a zero of the sum of a finite family of maximal monotone mappings in Banach spaces
Published in Optimization, 2022
Getahun B. Wega, Habtu Zegeye, Oganeditse A. Boikanyo
Some of the special cases of the Bregman Projection are the following: If, in (17), C is a closed and convex subset of a real reflexive Banach space E and , then the Bregman projection reduces to the generalized projection, which is defined by where , and J is the normalized duality mapping from E into .If, in (17), is a real Hilbert space and , then the Bregman projection reduces to the metric projection of x onto C.