query: What is the rank of matrix $pq^T +qp^T$?
positive_passages.docid: 86875
positive_passages.text: Finding the null space of symmetric matrix generated by outer product
negative_passages.docid: 27821
negative_passages.text: A problem on rank of $A$ and $\operatorname{adj}(A)$
negative_passages.docid: 45532
negative_passages.text: Let $A$ be a $n\times n$ matrix with entries $a_{ij}=i+j $ . Calculate rank of $A$
negative_passages.docid: 29788
negative_passages.text: Eigenvalues of a matrix whose square equals its transpose
negative_passages.docid: 7376
negative_passages.text: Find $\det(A^{2}+A^{T})$ when eigenvalues are $1,2,3$
negative_passages.docid: 48660
negative_passages.text: Proof that determinant rank equals row/column rank
negative_passages.docid: 33451
negative_passages.text: Sylvester rank inequality: $\operatorname{rank} A + \operatorname{rank}B \leq \operatorname{rank} AB + n$
negative_passages.docid: 149086
negative_passages.text: Determining $\mbox{tr}(A)$ and $\det(A)$ as functions of the eigenvalues of a matrix
negative_passages.docid: 42799
negative_passages.text: Let $x$ and $y$ be two non-zero $n × 1$ vectors. If $y^T$ denotes the transpose of $y$, what are the eigenvalues of the $n × n$ matrix $xy^T$
negative_passages.docid: 67058
negative_passages.text: Prove that $\text{rank}(X^TX)=\text{rank}(X)$
negative_passages.docid: 67990
negative_passages.text: Commutativity of matrix and its transpose
negative_passages.docid: 77137
negative_passages.text: The rank of Jacobian matrix at a point of affine variety is independent of choice of generators
negative_passages.docid: 121358
negative_passages.text: What is the exact relation between a full rank matrix and its determinant?
negative_passages.docid: 28565
negative_passages.text: Prove $\operatorname{rank}A^TA=\operatorname{rank}A$ for any $A\in M_{m \times n}$
negative_passages.docid: 123483
negative_passages.text: Prove that $\operatorname{rank}(A) + \operatorname{rank}(B) \ge \operatorname{rank}(A + B)$
negative_passages.docid: 69606
negative_passages.text: How to prove rank(A+) is no more than rank(A)?
negative_passages.docid: 82688
negative_passages.text: Least rank of a matrix