Questions tagged [traces]

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

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0answers
35 views

Necessary and sufficient conditions on kernels of trace-class operators

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 ...
0
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0answers
21 views

Loss function for matrix with fixed trace

I have a matrix $P \in \mathbb{R}^{n \times m}$ where $n \gg m$ and each row of $P$ has norm $1$. It is not hard to see that the $P^T P$ has a fixed trace $n$. I am try to find a loss function to push ...
3
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1answer
168 views

Frobenius algebras and traces of modules

$\newcommand{\Hom}{\mathscr{Hom}}$ 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 $C$, ...
0
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0answers
61 views

Derivative of trace uniformly bounded

I have that $\boldsymbol{\Omega{(\boldsymbol{\theta})}}$ is symmetric, positive definite, and uniformly bounded where $\boldsymbol{\theta} \in \boldsymbol{\Theta}$ with $\boldsymbol{\Theta}$ a ...
0
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0answers
100 views

Proof of Araki-Lieb-Thirring inequality

I would like to know a proof of Araki-Lieb-Thirring inequality. Let $H$ be a separable Hilbert space. Let $A$ and $B$ be positive and self-adjoint (not necessary bounded) operators on $H$. Let $BAB$ ...
3
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0answers
74 views

Is the secondary Euler characteristic a categorical trace?

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 ...
2
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1answer
414 views

Why trace is more natural than (preferred to) determinant for smooth map $f:M\to N$?

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 ...
5
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0answers
89 views

Closed paths, closed trails and traces

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 ...
5
votes
1answer
129 views

Trace inequality under consideration of definiteness

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} \...
10
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2answers
352 views

Inequality for trace of a symmetric product?

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 ...
4
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1answer
249 views

Closed paths, traces and spectra

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 ...
3
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1answer
126 views

Traces and closed walks that do not close before their time

Let $A$ be the adjacency matrix of a graph. Then, as is well-known and trivial to show, $\mathrm{Tr} A^k$ equals the number of closed walks of length $k$. Is there a similar way to express (a) the ...
5
votes
0answers
84 views

A bounded extension operator

Let $\Omega\subset\mathbb{R}^n$ be a bounded domain with smooth boundary $\partial\Omega$. Consider the harmonic extension operator $E :L^2(\partial \Omega) \rightarrow H^{1/2}(\Omega)$ which assigns ...
0
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0answers
152 views

Tensor contraction (vector-valued trace) on $\bigotimes_{i=1}^k\mathcal L(E_{i-1},E_i)$

If $E_i$ is a $\mathbb R$-vector space, then the vector-valued trace $\operatorname{tr}_{E_1}:(E_2\otimes E_1^\ast)\otimes(E_1\otimes E_0)\to E_1\otimes E_0$ (or tensor contraction) is the ...
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0answers
236 views

Computational complexity of computing the trace of a matrix product under some structure

I have two problems related to computing some trace, and some (possibly suboptimal) answers. My question is about a potential more efficient algorithm for each one. (More interested in an answer to ...
4
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0answers
286 views

Inequalities for trace/eigenvalues of product of multiple 2x2 matrices

Consider the matrix product $\prod_i^n A_i$, where each $A_i$ is a $2\times2$ matrix having the form $A_i = \left( \begin{smallmatrix} \lambda + \alpha_i & -\beta_i \\ 1 & 0\end{smallmatrix}\...
3
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0answers
77 views

Compatibility between the source and the boundary condition for an Helmholtz-type equation

Let $\Omega$ an open, convex, bounded domain in $\mathbb{R}^3$, and let us fix also $z\in\mathbb{C}\setminus\mathbb{R}$. Given $\phi\in H^{3/2}(\partial\Omega)$, I would like to show the existence of ...
7
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1answer
190 views

Hecke algebra relation versus $\operatorname{SL}_2$ trace relation

The quadratic relation in the (type $A$) Hecke algebra is $(T-t)(T+t^{-1}) =0$, which can be rewritten as $$ T-T^{-1} = t-t^{-1}$$ Suppose $A \in \operatorname{SL}_2(\mathbb{Q})$ with eigenvalues $a,...
4
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0answers
156 views

Why are traces an analogue to integrals?

In Poincare duality for singular cohomology, one integrates cohomology classes against a fundamental class to get a number $\int_{[M]} \omega$. In the formulation of Poincare duality in etale ...
0
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1answer
94 views

Trace of a finite hypercubic tensor

Is the trace of a finite hypercubic tensor defined? Clearly, for the bidimensional case $n \times n$ the trace is defined as the sum of the elements on the main diagonal: $$\operatorname {tr} (\...
1
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1answer
137 views

Why is the relative trace of Sobolev norms finite?

I am reading the 2009 Paper on Effective Equidistribution by Einsiedler, Margulis and Venkatesh (EMV). I do not understand Section 5.3 on the proof of (3.10). They want to prove that the relative ...
3
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1answer
359 views

Trace and exterior product

Let $V$ be a $2n$-dimensional complex vector space with base $\{e_1,\dotsc,e_n,f_1,\dotsc,f_n]\}$ Let $W \subset \wedge^n V$ be the subspace in the exterior product, with basis vectors $$ e_{i_1} \...
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0answers
155 views

Direct solution to maximum likelihood computation problem using the derivative of multivariate Gaussian w.r.t. covariance matrix

For an application, I need to compute the maximum loglikelihood of data coming from a $d$-dimensional multivariate Gaussian random variable: $$ \textbf{x} \sim \mathcal{N}(\boldsymbol{\mu}, \Sigma) $$ ...
2
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0answers
37 views

Nondegenerate linear maps functorially associated to algebras

In the sequel, "$k$-algebra" means "associative unital finite dimensional $k$-algebra. Apologies for the very long exposition. If $A$ is a $k$-algebra and $s:A\to k$ is $k$-linear, we say that $s$ ...
4
votes
2answers
999 views

Reconstruct a matrix from its traces

In my research I came across the following problem. Let $A$ be a symmetric and $\Gamma$ be a diagonal $n\times n$ matrices. The eigenvalues of $A$ are known $\lambda_1,\ldots\lambda_n$. The traces $\...
2
votes
0answers
108 views

Equality condition for Araki–Lieb–Thirring inequality

I'd like to have the equality condition in the Araki–Lieb–Thirring inequality $$\operatorname{Tr} [(BAB)^r]\leq \operatorname{Tr} [(B^{r}A^{r}B^{r})],$$ valid for $A,B$ semidefinite positive and $r\...
3
votes
0answers
110 views

Dixmier traces, Wodzicki residue and residues of zeta functions

Let $M$ be an $n$ dimensional closed manifold and consider an elliptic, pseudodifferential operator $P$ of order $-n$. Here are some facts which I had learned so far: 1. There exists a density defined ...
4
votes
1answer
174 views

Norm/trace of product inequality involving skew symmetric matrices

I wonder if the following inequality involving skew symmetric matrices is true: Suppose that $B,C \in \mathbb{R}^{d \times d}$ are skew-symmetric matrices, and $\Sigma \in \mathbb{R}^{d \times d}$ ...
5
votes
1answer
388 views

What is the trace of the integral operator $(\mathcal{L}f)(x)=\int_0^\infty (x \wedge y)f(y) \, d \pi(y)$?

Let $\pi$ denote a probability measure on $[0,+\infty)$ and let us assume that $$m:=\int_0^\infty x \, \mathrm{d} \, \pi(x)<+\infty.$$ Let us consider the integral operator $\mathcal{L}$ on $L_2(\...
7
votes
1answer
302 views

Is there a converse to the Brauer–Nesbitt theorem?

$\DeclareMathOperator\Tr{Tr}$Say that we have an algebra $R$ over $\mathbb{C}$. If, for two finitely generated (edit: and semisimple) $R$-modules $M, N$ we know that $\Tr_M(r)=\Tr_N(r)$ for all $r\in ...
4
votes
2answers
182 views

Bound on sum of $n$th super-diagonal entries in a $2n$ by $2n$ PSD matrix

Let $A,B,C\in\mathbb{R}^{n\times n}$ be such that $\left(\begin{array}{} A & B \\ B^T & C \end{array}\right)\succeq 0$. I would like to prove that $$\mathrm{trace}\,B \le \sum_{i=1}^n \sqrt{\...
1
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0answers
56 views

Is there a vector-valued trace such that $\text{tr}((L\otimes_π\text{id}_H)T)=LT$ for all $L∈\mathfrak L(H,\mathfrak L(H))$ and $T∈H\hat\otimes_πH$?

Let $H$ be a separable $\mathbb R$-Hilbert space $L\in\mathfrak L(H,\mathfrak L(H,\mathbb R))$ $T\in\mathfrak L(H)$ be nonnegative, self-adjoint and nuclear (trace-class) Note that$^1$ $$\...
1
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0answers
106 views

Traces in associative algebras

Are there some books or papers about the general definition of traces: If $\mathscr{A}$ is an associative algebra over $K$ then the space of traces is the set of all linear functionals $\tau:\mathscr{...
14
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5answers
2k views

Matrix trace & norm [closed]

For any nonnegative semidefinite matrix $A$ and any matrix $B$, we have $$\mbox{tr} (AB) \le \mbox{tr} (A) \, \|B\|$$ where $\mbox{tr}(\cdot)$ is the trace and $\|\cdot\|$ is the operator norm. How ...
13
votes
1answer
495 views

Trace-class operator satisfies $\sum |\lambda_n|<\infty$?

Here's an "exercise" which I thought should be easy, but which I find myself unable to do. Let $V$ be a Banach space. Recall that an operator $f:V\to V$ is trace-class if it is in the image of the ...
4
votes
1answer
288 views

Image of the trace map of ring of integers

Let $L/\mathbb{Q}$ be a finite Galois extension, and let $\mathcal{O}_L$ be the ring of integers of $L$. We have $tr_{L/\mathbb{Q}}(\mathcal{O}_L)=d\mathbb{Z}$ for some $d\geq 1.$ Fact. $d=1$ if ...
2
votes
2answers
176 views

A Characterization of the traces of functions in $W^{1,2}$

I have a question about the traces of functions in $W^{1,2}$. Let $D$ be a connected open subset of $\mathbb{R}^d$.We denote $W^{1,2}(D)$ by \begin{align*} W^{1,2}(D)=\{f \in L^{2}(D,dx) \mid \...
2
votes
1answer
99 views

Retractions for completely positive unital maps, and their effect on spectral diameter

Consider a non-singular, completely positive, unital map $\Psi: \mathbf M_k(\mathbb C) \to \mathbf M_h(\mathbb C)$. This map will have one or more retractions $\Phi: \mathbf M_h(\mathbb C) \to \mathbf ...
2
votes
1answer
131 views

Condition for non-vanishing trace

Let $A$ and $B$ be two full column rank real matrices of dimension $n \times m$, where $n \ge m$. Let $P$ be an $m\times m$ positive definite matrix. Question: Does there always exist a symmetric $n \...
11
votes
2answers
727 views

Trace of non-commutable matrices

Let $M_1$ and $M_2$ be two symmetric $d\times d$ matrices. What is the relationship between $tr(M_1M_2M_1M_2)$ and $tr(M_1^2 M_2^2 )$? P.S. I tried a few examples and found $$ tr(M_1M_2M_1M_2) \le tr(...
4
votes
1answer
170 views

Retractions for completely positive unital maps, with particularly nice norms

Consider a non-singular, completely positive, unital map $\Psi: \mathbf M_k(\mathbb C) \to \mathbf M_h(\mathbb C)$. This map will have one or more retractions. Does $\Psi$ admit a retraction $\Phi: \...
6
votes
1answer
286 views

Completely bounded norm for unital maps with completely positive sections

Consider a completely bounded unital map $\Phi: \mathbf M_h(\mathbb C) \to \mathbf M_k(\mathbb C)$. Suppose that $\Phi$ has right-inverse $\Psi$ which is completely positive. Is the operator norm of $\...
9
votes
0answers
207 views

Can we extend c.p. normal maps on a finite von Neumann algebra $M$ to $L_0(M)_+$?

Suppose that $M$ is a von Neumann algebra with a finite, normal, faithful trace $\tau$. Let $T\colon M\to M$ be a completely positive, normal map. Can $T$ be extended to a `positively linear map' ...
2
votes
1answer
303 views

Submodularity property of trace of inverse matrix

$\newcommand{\tr}{\operatorname{tr}}$Does submodularity property hold for the trace of a positive-definite hermitian matrix? I.e., does given any real symmetric positive-definite matrices $X,A,B$ $$ ...
3
votes
1answer
427 views

Norm and trace inequalities

If $A$ and $B$ are two positive definite matrices such that $\|A\| \leq \|B\|$ for every unitarily invariant norm $\| \cdot \|$, and $U$ is an $n\times k$ matrix with adjoint $V$ such that $VU = I_k$, ...
2
votes
1answer
241 views

Inequality on diagonal entries of a matrix product

Let $A$ and $B$ be two Hermitian matrices and let $D$ be a diagonal matrix. Does there exist any inequality involving the trace for the diagonal entries $(D A D A D A B)_{i,i}$? I am looking for ...
3
votes
2answers
2k views

Maximizing trace of $\mathrm V^T \mathrm A \mathrm V$ for $\mathrm A$ symmetric (alternate proof with min/max-theorem)

I'm trying to work out a proof for the following proposition: Let $A \in \mathbb{R}^{n,n}$ a real, symmetric matrix with eigenvalues $\lambda_1 \ge \lambda_2 \ge \cdots \ge \lambda_n$, then $$\max \...
1
vote
1answer
597 views

Trace 0 and Norm 1 elements in finite fields

Let $\mathbb{F}_{q^\ell}/\mathbb{F}_{q}$ be the extension of finite filed $\mathbb{F}_{q}$, where $\ell$ be a odd prime and $(\neq q)$. Take $\zeta\in\mathbb{F}_{q^\ell}$. Does there exist different $...
5
votes
0answers
144 views

Traces in finite extensions of integrally closed domains

$\def\fp{\mathfrak{p}}\def\fq{\mathfrak{q}}$I'm looking for a reference for the following commutative algebra fact. Let $A$ be an integrally closed integral domain, with field of fractions $K$. Let $...
2
votes
0answers
315 views

Trace of roots of unity has valuation more than 1 in uramified field

Let $F$ be a finite extension of $\mathbb{Q}_p$ (p is prime) and $K/F$ be a unramified extension of prime degree $\ell (\neq p)$. Denote $\mu_K$ be the group of roots of unity in $K.$ Does there exist ...