query: For non-negative definite symmetric matrices, $\mathrm{tr}(AB)\le \mathrm{tr}(A)\mathrm{tr}(B)$
positive_passages.docid: 155884
positive_passages.text: What are the conditions on $\text{tr}(AB) \leq \text{tr(A)} \text{tr(B)}$ to be true?
negative_passages.docid: 110271
negative_passages.text: Given a matrix with non-negative real entries, can you algebraically prove that it has a non-negative eigenvalue?
negative_passages.docid: 54211
negative_passages.text: If $A$ and $B$ are positive definite, then is $B^{-1} - A^{-1}$ positive semidefinite?
negative_passages.docid: 98631
negative_passages.text: Postive-semidefiniteness of matrix with entries $1/(a_i+a_j)$
negative_passages.docid: 95677
negative_passages.text: Linear Matrix Inequality of a matrix $T$ with $\| T \|_2 < \rho$
negative_passages.docid: 36775
negative_passages.text: Matrix semi-positive definite
negative_passages.docid: 114208
negative_passages.text: Does non-symmetric positive definite matrix have positive eigenvalues?
negative_passages.docid: 153372
negative_passages.text: Characterization of positive definite matrix with principal minors
negative_passages.docid: 125889
negative_passages.text: Determinant of rank-one perturbations of (invertible) matrices
negative_passages.docid: 151721
negative_passages.text: Uhlmann's Theorem: proof of $\text{tr}(A^{\dagger} B) = \langle m | A \otimes B |m\rangle $
negative_passages.docid: 111366
negative_passages.text: Does there exist a symmetric matrix $A$ such that $2^{\sqrt{n}}\le |\operatorname{Tr}(A^n)|\le2020 \ \cdot 2^{\sqrt{n}}$ for all $n$
negative_passages.docid: 84522
negative_passages.text: A better proof for $\det(P) = \pm1$ if $P$ is an orthogonal matrix
negative_passages.docid: 87262
negative_passages.text: How to prove positive semidefiniteness of two matrices through Schur Complement?
negative_passages.docid: 57318
negative_passages.text: Property of the conjugate transpose matrix with inner product
negative_passages.docid: 52882
negative_passages.text: find nonsingular matrices $B$ and $C$ that satisfy $BC+ CB= 0$
negative_passages.docid: 70381
negative_passages.text: Find the determinant of a symmetric matrix
negative_passages.docid: 85550
negative_passages.text: Positive definiteness of difference of inverse matrices