There is an algorithm that for any input matrix $A \in \mathbb{R}^n$ satisfies $x^\top A x>0$ for all $x \in \mathbb{R}^n$, e.g. by using Cholesky algorithm. Is there an algorithm that, for matrix $A \in \mathbb{R}^n$ and a subspace $V \subseteq \mathbb{R}^n$, check if $x^\top A x >0$ for all $x \in V \setminus \{0\}$?
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$\begingroup$ Oops you are right, thank you! Question edited. $\endgroup$– nivotkoCommented Apr 23, 2019 at 8:29
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Of course. Use Gram-Schmidt to construct an orthonormal basis $\{u_i\}$ of $V$, and use your algorithm on the matrix with entries $u_i^\top A u_j$.
answered Apr 18, 2019 at 11:54
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$\begingroup$ Do you mean Cholesky algorithm on matrix with entries $u_i^\top A u_j$? $\endgroup$– nivotkoCommented Apr 23, 2019 at 8:30
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$\begingroup$ Whichever algorithm you want to use for checking positive definiteness. $\endgroup$ Commented Apr 23, 2019 at 11:39
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$\begingroup$ Great, I will try it. Thank you very much! :) $\endgroup$– nivotkoCommented May 3, 2019 at 14:49
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$\begingroup$ Is it important for the basis to be orthonormal? $\endgroup$ Commented Jul 29, 2023 at 21:39