query: Let $A$ and $B$ be positive definite matrices and $S_{M}= \{x:x^TM^{-1}x \leq 1\}$ . Show that $S_{A} \subseteq S_{B}$ iff $B \succeq A$
positive_passages.docid: 139011
positive_passages.text: How to prove $B-A \succeq 0 \Leftrightarrow$ ellipsoid $x^TBx \leq 1$ contains $x^TAx \leq 1$?
negative_passages.docid: 64955
negative_passages.text: Show that there are $ a,b \geq 0 $ so that $ |f(x)| \leq ax+b, \forall x \geq 0.$
negative_passages.docid: 95677
negative_passages.text: Linear Matrix Inequality of a matrix $T$ with $\| T \|_2 < \rho$
negative_passages.docid: 1478
negative_passages.text: An inequality for powers of reals: $a^{1/n}-b^{1/n}<(a-b)^{1/n}$
negative_passages.docid: 29198
negative_passages.text: How to prove the following inequality $|\prod_{i=1}^{i=n}a_i-\prod_{i=1}^{i=n}b_i| < n\delta$?
negative_passages.docid: 87425
negative_passages.text: Prove that if $A$ is a square matrix with integer entries and $\det(A) = \pm 1$, then $A^{-1}$ contains all integer entries.
negative_passages.docid: 135171
negative_passages.text: Under what conditions is the difference between a rank $1$, postive semidefinite matrix and a hermitian matrix positive semidefinite?
negative_passages.docid: 98631
negative_passages.text: Postive-semidefiniteness of matrix with entries $1/(a_i+a_j)$
negative_passages.docid: 136789
negative_passages.text: How to prove $a \lt b$ for all $a \in A, b \in B$ implies $\sup A \leq \inf B$?
negative_passages.docid: 153372
negative_passages.text: Characterization of positive definite matrix with principal minors
negative_passages.docid: 67820
negative_passages.text: Showing that $(A_{ij})=\left(\frac1{1+x_i+x_j}\right)$ is positive semidefinite
negative_passages.docid: 143130
negative_passages.text: Proving that the supremum of $(a,b)$ is $b$.
negative_passages.docid: 68132
negative_passages.text: Small positive eigenvalues found for a negative definite matrix
negative_passages.docid: 27306
negative_passages.text: Positive semi-definite vs positive definite
negative_passages.docid: 130498
negative_passages.text: Linear algebra proof regarding matrices
negative_passages.docid: 25488
negative_passages.text: Let $A,B$ be $m\times m$ matrices such that $AB$ is invertible. Show $A,B$ invertible.
negative_passages.docid: 2798
negative_passages.text: A and B are nxn matrices, $A =B^TB$. Prove that if rank(B)=n, A is pos def, and if rank(B)<b, A is pos semi-def.