query: Relation of $det(AA^T)$ and $det(A^TA)$ with Range and Nullspaces of a Matrix
positive_passages.docid: 78307
positive_passages.text: Under what conditions is $AA^T$ invertible?
negative_passages.docid: 5058
negative_passages.text: Eigenvalues of the rank one matrix $uv^T$
negative_passages.docid: 125030
negative_passages.text: Prove that $\det(A)=p_1p_2-ba={bf(a)-af(b)\over b-a}$
negative_passages.docid: 3682
negative_passages.text: If $V = \text{null}(\textsf{T}-\lambda\textsf{I}) \oplus \text{range}(\textsf{T}-\lambda\textsf{I})$, then prove that $\textsf{T}$ is diagonalizable
negative_passages.docid: 79276
negative_passages.text: Find $\min(\operatorname{trace}(AA^T))$ for invertible $A_{n\times n}$
negative_passages.docid: 10083
negative_passages.text: Determinant of matrix times a constant.
negative_passages.docid: 17394
negative_passages.text: Sign of determinant when using $det A^\top A$
negative_passages.docid: 125348
negative_passages.text: Does $\det(A + B) = \det(A) + \det(B)$ hold?
negative_passages.docid: 60598
negative_passages.text: Matrix $A^T A$ as sum of outer products
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: 22149
negative_passages.text: How to prove $dim(Ker (A- \lambda I))=dim (Ker(A^t- \lambda I))$?
negative_passages.docid: 78224
negative_passages.text: Suppose $A$ is a general $n \times n$ matrix and $B$ is obtained by interchanging two rows of $A$. Prove that $\det(B) = -\det(A)$
negative_passages.docid: 40517
negative_passages.text: Determinant of a non-square matrix
negative_passages.docid: 18670
negative_passages.text: What is the gradient of $x^T A\, x$ with respect to the matrix $A$?
negative_passages.docid: 86875
negative_passages.text: Finding the null space of symmetric matrix generated by outer product
negative_passages.docid: 75855
negative_passages.text: Show that $\det{(A + sxy^*)}$ is linear in $s$
negative_passages.docid: 65305
negative_passages.text: Minimal polynomial for an invertible matrix and its determinant