query: rank($A$)=rank($A^T$)
positive_passages.docid: 19203
positive_passages.text: Is the rank of a matrix the same of its transpose? If yes, how can I prove it?
negative_passages.docid: 155579
negative_passages.text: Show that $End_A(A)$ = {$r_a$ | $a ∈ A$}
negative_passages.docid: 6905
negative_passages.text: Rank of product of a matrix and its transpose
negative_passages.docid: 45469
negative_passages.text: If $a_k = \operatorname{rank}(S^k)$, deduce that $a_k - a_{k+1} \geq a_{k+1} - a_{k+2}$
negative_passages.docid: 50384
negative_passages.text: A matrix and its transpose have the same set of eigenvalues/other version: $A$ and $A^T$ have the same spectrum
negative_passages.docid: 46128
negative_passages.text: Representation of Matrix with Rank 1
negative_passages.docid: 37162
negative_passages.text: Prove that matrices have equal rank.
negative_passages.docid: 134402
negative_passages.text: $\mathrm{rank}(A)+\mathrm{rank}(I-A)=n$ for $A$ idempotent matrix
negative_passages.docid: 121221
negative_passages.text: Rank-Nullity Theorem
negative_passages.docid: 67058
negative_passages.text: Prove that $\text{rank}(X^TX)=\text{rank}(X)$
negative_passages.docid: 27248
negative_passages.text: If $A$ is an $m \times n$ matrix, prove that $Ax=0$ has infinite solutions $\iff$ $\text{rank}(A)<n$
negative_passages.docid: 11895
negative_passages.text: If $AB=0$ prove that $\mathrm{rank}(A)+\mathrm{rank}(B)\leq n$
negative_passages.docid: 48660
negative_passages.text: Proof that determinant rank equals row/column rank
negative_passages.docid: 156877
negative_passages.text: Rank-one perturbation proof
negative_passages.docid: 4952
negative_passages.text: Let $A$ be a diagonalizable matrix and $\lambda$ be an eigenvalue of A. Prove that rank($\lambda I - A$) = rank(($\lambda I - A)^2$).
negative_passages.docid: 75117
negative_passages.text: $\mathrm{rank}(AB-BA)=1$ implies $A$ and $B$ are simultaneously triangularisable
negative_passages.docid: 23224
negative_passages.text: $\mbox{rank}(A)=1$ implies $\det(A+I)=\mbox{trace}(A)+1$