query: Proof that $A=\lambda U$
positive_passages.docid: 145885
positive_passages.text: Linear transformation preserving orthogonality
negative_passages.docid: 27564
negative_passages.text: Prove that $\|UVU^{-1}V^{-1}-I\|\leq 2\|U-I\|\|V-I\|$
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: 90332
negative_passages.text: Prove that that $U(n)$ is an abelian group.
negative_passages.docid: 100968
negative_passages.text: Why is $|e^{i \lambda z}| |e^{- \lambda y}|= |e^{- \lambda y}|$ here?
negative_passages.docid: 62887
negative_passages.text: Proof that a linear transformation is continuous
negative_passages.docid: 72046
negative_passages.text: Prove all eigenvalues of $A^*A$ are non-negative
negative_passages.docid: 92291
negative_passages.text: $U_1, U_2, ..., U_k$ subspaces of linear space $X$ with same dimensions, prove $X=U_1\oplus V=U_2\oplus V=...=U_k\oplus V$
negative_passages.docid: 121802
negative_passages.text: Prove that every self-adjoint unitary linear operator can be expressed in the form $U\alpha = \beta - \gamma$
negative_passages.docid: 149664
negative_passages.text: Norm is sup of inner products (proof).
negative_passages.docid: 34147
negative_passages.text: $\|u\|\leq \|u+av\| \Longrightarrow \langle u,v\rangle=0$
negative_passages.docid: 142229
negative_passages.text: Prove that if $V \cap \text{Span}\{u_1, u_2,...,u_k\} = \{ 0\} $, then $Au_1, Au_2,...,Au_k$ are linearly independent
negative_passages.docid: 133884
negative_passages.text: A proof using Yoneda lemma
negative_passages.docid: 14576
negative_passages.text: $(U\cap W)^0 = U^0 + W^0$
negative_passages.docid: 7164
negative_passages.text: Proof of Darboux's theorem
negative_passages.docid: 117688
negative_passages.text: Prove that $\def\Aut{\operatorname{Aut}}\Aut(\mathbf{Z_{n}})\simeq \mathbf{Z_{n}^{*}}$
negative_passages.docid: 98662
negative_passages.text: $P(|X|\ge\lambda a)\ge (1-\lambda)^2a^2$ for $0\le \lambda \le 1$