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Results tagged with quadratic-forms
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user 8799
Algebraic and geometric theory of quadratic forms and symmetric bilinear forms, e.g., values attained by quadratic forms, isotropic subspaces, the Witt ring, invariants of quadratic forms, the discriminant and Clifford algebra of a quadratic form, Pfister forms, automorphisms of quadratic forms.
21
votes
The statement that $A \ge B$ implies $A^{-1} \le B^{-1}$ is still true for matrices?
This is a well-known fact. A simple proof : setting $y=B^{1/2}x$, we have $\|y\|^2\le y^TB^{-1/2}AB^{-1/2}y$, that is $I_n\le B^{-1/2}AB^{-1/2}$. The eigenvalues of the latter symmetric matrix are thu …
- 50.6k
6
votes
Characterising semi-definite positiveness on vectors with non-negative entries
(After Noah Stein's answer) By definition, the dual cone $C^\star$ is spanned by matrices $v\otimes v$ with $v\ge0$. The following counter-example is due to Hall. The $5\times5$ symmetric matrix
$$S=\ …
- 50.6k
4
votes
Positive quadratic polynomial
This is a bit too long for a comment ...
Forty years ago, I wrote a paper Formes quadratiques et calcul des variations, published in J. Maths. Pures & Appl. 62 (1983), p 177-196. It dealt with such a …
- 50.6k