What is semipositive definite?

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SUMMARY

The term "semipositive definite" refers to a bilinear real form where the condition ##\beta(v,v) \geq 0## holds true for all vectors ##v## in ##\mathbb{R}^n##. This concept is often synonymous with "positive semidefinite," as highlighted in Jackson's 3rd edition, Chapter 1, page 44. The key distinction lies in the existence of non-zero vectors ##v \neq 0## for which ##\beta(v,v) = 0##, differentiating it from a positive definite inner product.

PREREQUISITES
  • Understanding of bilinear forms in linear algebra
  • Familiarity with the concepts of positive definite and positive semidefinite matrices
  • Basic knowledge of inner product spaces
  • Proficiency in mathematical notation and real analysis
NEXT STEPS
  • Study the properties of positive semidefinite matrices in linear algebra
  • Explore the implications of bilinear forms in functional analysis
  • Learn about the applications of semipositive definite forms in optimization problems
  • Investigate the differences between positive definite and semipositive definite forms
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Mathematicians, students of linear algebra, and anyone studying real analysis or optimization theory will benefit from this discussion on semipositive definite forms.

anbhadane
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In Jackson,(3rd edition) Chapter 1 , page no, 44 He uses the word "semipositive definite" what is it? is it "non-negative" definite?
 
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It is usually called positive semidefinite unless Jackson didn't use something completely different. It means that for a bilinear real form: ##\beta\, : \,\mathbb{R}^n\times \mathbb{R}^n\longrightarrow \mathbb{R}## we have ##\beta(v,v)=\langle v,v\rangle \geq 0##.

The essential difference to a usual inner product (positive definite) is, that there may be vectors ##v\neq 0## such that ##\beta(v,v)=0##.
 
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Oh, Thank you, got it.
 

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