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For questions involving the trace of a square matrix, i.e. the sum of the elements on the main diagonal.

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Let $P \in \mathbb{R}^{d \times d}$ be an rank $p < d$ orthogonal projection matrix given by $P = VV^T$, $V \in \mathbb{R}^{d \times p}$. Do we have that $$ \underset{P^2 = P,\; \text{rank}(P) = p}{...
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Suppose $U$ is an open bounded set in $\mathbb{R}^n$ ($n \geq 3)$ with Lipschitz boundary. Let $f \in H^1(U)$ with $f>c$ a.e in $U$ for some $c \in \mathbb{R}$. Suppose for any $\epsilon >0$ ...
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In this paper of Olivier & Żurek, the notion of quantum discord is introduced. I already asked one question related to this paper in this discussion. However, I realized that one (possibly small) ...
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Let $Z_1$ and $Z_2$ be two independent standard normal random variables and $A_1$,$A_2$ two commuting matrices. Suppose that $B$ does not commute with either of them. What tools does one have if one ...
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Let $G$ be a finite gorup. I think the orthogonality relation between elements of $\mathbb{C}$-irreducible representation of $G$ is well-known. If $R$ is any orthogonal realization of an irreducible ...
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For the Hilbert space $H^N:=L((\mathbb R^{3})^N,\mathbb C)$, consider the projection operator $D: H^N\to H^N$ as follows : $$D(\Phi):=\left(\int_{(\mathbb R^{3})^N}\overline{\Psi(x_1,\ldots, x_N)}\Phi(...
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Let $V$ be a finite dimensional real / complex vector space and consider the space $L(V,V)$ of linear operators on $V$. Fix $n \in \mathbb{N}$ and let $\mathcal{M}$ be the real / complex vector space ...
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Let $G$ be a measured étale groupoid with quasi-invariant measure $\mu$ (that induces the reduced $C^* $-algebra, meaning it has full support) with associated equivalent measures $\nu,\nu^{-1}$. Is ...
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I'm cross-posting this question from Math.SE, as it didn't get much attention there. Let $H$ be a separable Hilbert space and $T \in L(H)$ a trace-class operator. It is well known that the trace of $T$...
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This question is really an addendum to Geometric interpretation of trace There is a nice account of the trace in Chris Doran's thesis here: http://geometry.mrao.cam.ac.uk/wp-content/uploads/2015/02/...
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Let $A$ be an $N\times N$ Hermitian matrix. For $p+q$ even, consider the following inequality: $$\frac{1}{N}\sum_{i=1}^N (A^p)_{ii} (A^q)_{ii} \geq \Big(\frac{1}{N}\sum_{i=1}^N (A^p)_{ii} \Big) \Big(\...
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Let $G := G(k,V)$ denote the Grassmannian of $k$-linear subspaces in a $\mathbb{C}$-vector space $V$ of dimension $n$. Let $S$ denote the tautological bundle over $G$. There is a canonical map on ...
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Let $P \in \mathbb{R}^{d \times d}$ be the transition kernel for a Markov chain with stationary measure $\pi$ and define $P^\ast$ to be the time-reversed transition kernel defined by $P^\ast_{ij} := ...
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Let $R$ be a (possibly non-commutative) unital ring and $M$ be a left $R$-module. If $M$ is finitely generated and projective, the natural map $$\iota:\mathrm{Hom}_R(M,R)\otimes_R M\to \mathrm{Hom}_R(...
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Let $h,m\in[1,\infty)$. I would like to verify that the following sum diverges logarithmically \begin{equation} \sum_{d=0}^{\infty} \frac{2d+1}{2h^2(1+\frac{d(d+1)}{h^2})(1+\frac{d(d+1)}{h^2m ^2})^{2}}...
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$$Tr(A|A^T U \Sigma U^T|) \leq Tr(AA^T U \Sigma U^T)$$ where $A \in \mathbb{R}^{m\times n}$ is a real rectangular matrix, $U \in \mathbb{R}^{n \times n}$ is a orthonormal matrix and $\Sigma$ is a ...
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Problem. Let $n_1,\ldots,n_s,m_1,\ldots,m_s\ge 0$ be nonnegative integers and set $m := \sum_{i=1}^s m_i$ and $n := \sum_{i=1}^s n_i$. Let $\oplus$ be an operation on matrices which stacks them in a ...
eepperly16's user avatar
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Let $\texttt{R}$ be an $\texttt{E}$-infinity ring and let $\texttt{M,N}$ be $\texttt{E}$-infinity modules. Under what conditions do we have $$ \texttt{[M, N] ≅ [M,R] ⊗ N}$$ Under ordinary ...
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Suppose that I have $n$ matrices $A_1, \ldots, A_n \in \mathbb{R}^{m \times m}$ with $m \gg n$. Can I find $n$ new matrices $B_1, \ldots, B_n \in \mathbb{R}^{n \times n}$ that have the same 3-way ...
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Suppose $A$ is a positive definite matrix and we can bound its trace as $l \le \operatorname{tr} (A) \le L$. I am wondering if it is possible to find the upper and lower bounds on the trace of $A^{-1}$...
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Let $A, B, C \in \mathrm{GL}(2,\mathbb{C})$ be reflections (i.e., their eigenvalues are $\pm 1$). Please show that $$ \DeclareMathOperator\Tr{Tr}\{\Tr(AB)\}^2+\{\Tr(BC)\}^2 + \{\Tr(CA)\}^2 - \{\Tr(AB)\...
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I am working on a following problem in my free time (which is a simplified version of a problem described here - arxiv.org/abs/0711.2613): Let $A$, $B$ be zero-trace $4 \times 4$ matrices that meet ...
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I've seen a host of results concerning computations for $$\mathbb{E} \left[ \operatorname{tr} A^{i_1}\cdots \operatorname{tr} A^{i_j} \,\overline{\operatorname{tr} A^{k_1} \cdots \operatorname{tr} A^{...
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Let $G$ be a finitely generated group with a bound on its complex unitary irreducible representations: That is assume all complex unitary irreducibles of $G$ have degrees at most $k$ for some integer $...
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$\DeclareMathOperator\Tr{Tr}$I have a problem with understanding the proof of Proposition 6.8 in the book ,,Elements of Noncommutative Geometry''. One can find the formulation of this proposition here ...
truebaran's user avatar
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Let $A$ and $B$ be positive-definite matrices of the same size. For any $t \ge 0$, define $$ u(t) := \mbox{trace}(A^{-1/2}e^{-tB} (AB+BA) e^{-tB}A^{-1/2}). $$ Question. What are necessary and ...
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There is a theorem of Procesi that the ring of polynomial functions on tuples $(A_1,A_2, \dots, A_m)$ of $n \times n$ matrices, which are invariant under simultaneous conjugation, is generated by ...
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For matrices $A$ it is well known that the spectrum is invariant under transpose $\sigma(A^T) = \sigma(A)$. Furthermore, the spectrum of the adjoint matrix $\sigma(A^*) = \overline{ \sigma(A)}$ the ...
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I found a problem in my textbook and I have tried solving it, but I had no succes. The problem is: Let $A$ and $B$ be $n \times n$ matrices with complex number entries. Given that $AB−BA$ is ...
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I am reading the article On existence of weak solutions of the Navier-Stokes equations in regions with moving boundaries from Fujita and Kato and at some point they use an argument I have some trouble ...
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$\DeclareMathOperator\tr{tr}$Suppose we have positive semidefinite matrices $A_1, \dotsc, A_n$ and $B_1, \dotsc, B_n$ of the same dimension. Do we have a Hölder inequality for the trace of the ...
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$\DeclareMathOperator{\tr}{tr}$ Fix an integer $n \geq 2$. Let $A_1, \dotsc, A_n$ be hermitian positive semidefinite matrices, with each $A_i$ being $m$ by $m$. Consider the symmetric group $S_n$ on $...
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Let say I have two $n$ x $n$ matrices $A$ and $B$ where all elements are real positive values. I want to find some $n$ x $n$ permutation matrix $P$ such that $\operatorname{tr}(P A P ^T B)$ is ...
Chase Roberts's user avatar
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Let $V=V^0\oplus V^1$ be a super vector space (https://en.wikipedia.org/wiki/Super_vector_space) Is there a special definition of an inner product on $V$ other than just an inner product on the ...
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$\DeclareMathOperator\THH{THH}\DeclareMathOperator\Perf{Perf}\DeclareMathOperator\map{map}\DeclareMathOperator\tr{tr}\DeclareMathOperator\HH{HH}\DeclareMathOperator\fib{fib}\DeclareMathOperator\id{id}\...
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Note that this question was already posted on MSE: https://math.stackexchange.com/questions/4290741/tensor-product-of-operator-subalgebras-and-properties-of-the-trace Let $V$ be a vector space and let ...
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Let $\mathfrak{g}$ be a simple Lie algebra. Let $\rho$ be a representation of $\mathfrak{g}$ on a finite-dimensional vector space $E$. Consider now the bilinear form on $\mathfrak{g}$: \begin{equation}...
Peter's user avatar
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Suppose that $T \in TC(l^2( \mathbb{Z}))$ is trace class. Consider its kernel $ T(i,j) = \langle e_i, T e_j \rangle $ where $ \{e_i\}_{i \in \mathbb{Z}}$ is an ONB for $l^2( \mathbb{Z})$. Now, ...
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Let $\Omega$ be a bounded open set with smooth boundary and $E$ a set of finite perimeter in $\Omega$, i.e. $$P(E;\Omega)=\left\{\int_E\text{div}\: T\:dx:T\in C^\infty_c(\Omega;\mathbb{R}^n), |T|\leq1\...
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Crossposted from math.SE Let $\mathbb P_n$ be the space of all $n \times n$ self-adjoint positive definite matrices. Consider the function $\varphi: \mathbb P_n \longrightarrow \mathbb R$ defined by $...
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Let $A$ be an $n \times n$ random symmetric matrix with $E(A_{i j}) = 0$, $Var(A_{i j}) = 1/n$ for any $i,j$. $B$ is an $n \times n$ symmetric matrix with $B_{ii} = 0$. I need to find a upper bound of ...
Doris's user avatar
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1 answer
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Let $\mathcal{H}$ be a separable Hilbert space and let $TC( \mathcal{H})$, $HS(\mathcal{H})$ be the space of trace-class operators and Hilbert-Schmidt operators on $\mathcal{H}$. Recall that these ...
Frederik Ravn Klausen's user avatar
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2 votes
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Question: Let $K \in L^2(R^n\times R^n)$. Are "explicit" necessary and sufficient conditions known such that $K$ is the kernel of some trace-class operator $A \in TC(L^2(R^n))$? We know that ...
Nemis L.'s user avatar
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$\newcommand{\Hom}{\mathscr{Hom}}$ Let $C$ be a cocomplete closed symmetric monoidal category, and the tensor product preserves colimits in each variable; Let $A$ be a commutative algebra in $C$, ...
Maxime Ramzi's user avatar
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Context: The ordinary Euler characteristic of a complex (satisfying appropriate finiteness conditions so that all cohomology groups are finite-dimensional over some field ''k'', say, and only finitely ...
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Cross-post from MSE. For a continuous map $f:(M,g)\to (N,h)$, between Riemannian manifolds $(M,g)$ and $(N,h)$ we can pullback $h$ by $f$. Most experts take the trace from this new tensor and work ...
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Let $A$ be the adjacency matrix of a (non-oriented) graph $\Gamma$. Then $\textrm{Tr} A^k$ equals both the sum $\sum_i \lambda_i^k$ of $k$th powers of eigenvalues of $A$, on the one hand, and the ...
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Let $G \in \mathbb{R}^{3 \times 3}$ a symmetric, but indefinite matrix and $U \in \mathbb{R}^{3\times 3}$ a symmetric and positive definite matrix. I would like to prove the inequality $$ \text{Tr} \...
TARS's user avatar
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Let $A$ be a real, positive-definite, symmetric operator on an $n$-dimensional space $V$. Write $\odot^k A$ for the action of $A$ on the symmetric power $\odot^k V$. Let $v_1,\dotsc,v_n$ be a basis ...
H A Helfgott's user avatar
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Let $\Gamma$ be a graph. Write $A$ for its adjacency matrix. It is trivial to show that $\mathrm{Tr} A^k$ equals the number of closed walks of length k, that is, the number of $k$-step walks that ...
H A Helfgott's user avatar
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