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Then ρ ( A ) is a simple eigenvalue of A, associated to a positive eigenvector, where ρ ( A ) denotes the spectral radius of A. Moreover, ρ ( A ) > 0. Lemma 2.4 [12.4
B is called the Sylvester matrix of A associated with A.
Lemma 2.4 Let A be a K-matrix, B be the Sylvester matrix of A associated with A , then B is a K-matrix.
The criterion is given in terms of a certain algebraic hypersurface S π in the maximal ideal of A associated to a linear projection π on with range Soc A), where we assume that dimk A>1.
Note that, if θ ( x ) = 1 2 x 2, then the K-constrained generalized pseudoinverse A K, θ † of A associated with θ reduces to the K-constrained pseudoinverse A K † of A in Definition 4.1.
Moreover, X has a completely orthonormal base formed by eigenfunctions of A associated with the eigenvalues -n2, which is given x n = 2 π sin ( n ξ ), n ∈ ℕ.
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Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com