query: $A$ is normal if and only if $trace(A^\ast A)=\sum_{i=1}^n|\lambda_i|^2.$ where the $\lambda_i$ are the eigenvalues of $A$.
positive_passages.docid: 153448
positive_passages.text: Show that $A\in\mathbb{C}_n$ is normal $\iff$ $tr(A*A) = \sum_{i = 1}^n|\lambda_i|^2$, where $\lambda_1,...,\lambda_n$ are the eigenvalues of $A$.
negative_passages.docid: 67637
negative_passages.text: How come $\lambda(\mathbb{Q})= \inf\left\{\sum_{n \ge 1} \lambda([a_n,b_n)) : \mathbb{Q} \subset \bigcup_{n \ge 1} [a_n,b_n)\right\} = 0$?
negative_passages.docid: 117533
negative_passages.text: $|A-\lambda I|$ and $|\lambda I-A|$
negative_passages.docid: 23224
negative_passages.text: $\mbox{rank}(A)=1$ implies $\det(A+I)=\mbox{trace}(A)+1$
negative_passages.docid: 56943
negative_passages.text: Normal operator with real spectrum is hermitian.
negative_passages.docid: 71819
negative_passages.text: Prove that $H$ is normal.
negative_passages.docid: 73254
negative_passages.text: spectral radius in normed space is subadditive.
negative_passages.docid: 101467
negative_passages.text: Let $T:V\to V$ be a linear map, where $V$ is an inner product space. Given $T^2=\frac{T+T^*}{2}$, prove that $T$ is normal and that $T=T^2$.
negative_passages.docid: 7376
negative_passages.text: Find $\det(A^{2}+A^{T})$ when eigenvalues are $1,2,3$
negative_passages.docid: 54099
negative_passages.text: What are the eigenvalues of $\operatorname{ad}x$?
negative_passages.docid: 109675
negative_passages.text: Finite sum of eigenspaces (with distinct eigenvalues) is a direct sum
negative_passages.docid: 134981
negative_passages.text: Eigenvalues of $L^\dagger L$ are positive
negative_passages.docid: 104036
negative_passages.text: Right eigenvector λ and left eigenvector are orthogonal.
negative_passages.docid: 149664
negative_passages.text: Norm is sup of inner products (proof).
negative_passages.docid: 106097
negative_passages.text: Spectrum of the sum of two commuting matrices
negative_passages.docid: 73952
negative_passages.text: Upper bound for the sum of absolute values of the eigenvalues
negative_passages.docid: 110086
negative_passages.text: Let $A$ be an $m \times n$ matrix. Show $A^TA$ and $AA^T$ have the same eigenvalues