Abstract
Let V be an inner product space. If α is a self-adjoint endomorphism of V and v ∈ V then (math) and so (α(v),v) ∈ ℝ. An endomorphism a of V is positive if and only if it is self-adjoint and satisfies the condition that 0 (α(v),v) ∈ ℝ for all 0 V ≠ v ∈ V. Thus, for example, σ1 ∈ End(V) is positive. We note immediately that if α ∈ End(V) is positive then α must be a monomorphism since α(v) = 0 v implies that (α(v),v) = 0 and so we must have v = 0 v .
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© 1995 Springer Science+Business Media Dordrecht
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Golan, J.S. (1995). Endomorphisms of Inner Product Spaces. In: Foundations of Linear Algebra. Kluwer Texts in the Mathematical Sciences, vol 11. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-8502-6_15
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DOI: https://doi.org/10.1007/978-94-015-8502-6_15
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-4592-8
Online ISBN: 978-94-015-8502-6
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