All Questions
139 questions
0
votes
0
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62
views
An inequality regarding operator concave function
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 $...
- 141
8
votes
1
answer
249
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,...
- 2,869
35
votes
2
answers
9k
views
tr(ab) = tr(ba)?
It is well known that given two Hilbert-Schmidt operators $a$ and $b$ on a Hilbert space $H$, their product is trace class and $tr(ab)=tr(ba)$. A similar result holds for $a$ bounded and $b$ trace ...
- 43.2k
1
vote
0
answers
1k
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 ...
- 123
7
votes
2
answers
788
views
An extension of the Golden-Thompson inequality
For three symmetric positive semidefinite matrices $A, B,C$, I am trying to figure out if the following inequality holds, at least in some cases:
$$ \operatorname{tr} \left( A e^{B+C} \right) \leq \...
- 385
3
votes
1
answer
163
views
Traces and closed walks that do not close before their time
Let $A$ be the adjacency matrix of a graph. Then, as is well-known and trivial to show, $\mathrm{Tr} A^k$ equals the number of closed walks of length $k$.
Is there a similar way to express (a) the ...
- 20.2k
4
votes
0
answers
457
views
Inequalities for trace/eigenvalues of product of multiple 2x2 matrices
Consider the matrix product $\prod_i^n A_i$,
where each $A_i$ is a $2\times2$ matrix having the form $A_i = \left( \begin{smallmatrix} \lambda + \alpha_i & -\beta_i \\ 1 & 0\end{smallmatrix}\...
- 695
2
votes
0
answers
267
views
Necessary and sufficient conditions on kernels of trace-class operators
Question: Let $K \in L^2(R^n\times R^n)$. Are "explicit" necessary and sufficient conditions known such that $K$ is the kernel of some trace-class operator $A \in TC(L^2(R^n))$?
We know that ...
- 143
15
votes
1
answer
4k
views
trace(xy)=trace(yx) in full generality
It is well known that, for square matrix $x$ and $y$, we have $\operatorname{tr}(xy)=\operatorname{tr}(yx)$. Here of course the trace of a matrix is just the sum of the elements of the diagonal.
The ...
- 3,704
3
votes
0
answers
145
views
Is the secondary Euler characteristic a categorical trace?
Context: The ordinary Euler characteristic of a complex (satisfying appropriate finiteness conditions so that all cohomology groups are finite-dimensional over some field ''k'', say, and only finitely ...
- 2,040
3
votes
1
answer
814
views
Trace and exterior product
Let $V$ be a $2n$-dimensional complex vector space with base $\{e_1,\dotsc,e_n,f_1,\dotsc,f_n]\}$ Let $W \subset \wedge^n V$ be the subspace in the exterior product,
with basis vectors
$$
e_{i_1} \...
- 15.8k
5
votes
0
answers
112
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 ...
- 20.2k
7
votes
1
answer
349
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 ...
- 463
5
votes
0
answers
101
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 ...
- 81
13
votes
1
answer
724
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 ...
- 43.2k
4
votes
0
answers
321
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 ...
- 4,164
4
votes
2
answers
222
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
$$\mathrm{trace}\,B \le \sum_{i=1}^n \sqrt{\...
- 392
5
votes
1
answer
710
views
What is the trace of the integral operator $(\mathcal{L}f)(x)=\int_0^\infty (x \wedge y)f(y) \, d \pi(y)$?
Let $\pi$ denote a probability measure on $[0,+\infty)$ and let us assume that
$$m:=\int_0^\infty x \, \mathrm{d} \, \pi(x)<+\infty.$$
Let us consider the integral operator $\mathcal{L}$ on $L_2(\...
- 51
3
votes
0
answers
82
views
Compatibility between the source and the boundary condition for an Helmholtz-type equation
Let $\Omega$ an open, convex, bounded domain in $\mathbb{R}^3$, and let us fix also $z\in\mathbb{C}\setminus\mathbb{R}$. Given $\phi\in H^{3/2}(\partial\Omega)$, I would like to show the existence of ...
- 943
1
vote
1
answer
162
views
Why is the relative trace of Sobolev norms finite?
I am reading the 2009 Paper on Effective Equidistribution by Einsiedler, Margulis and Venkatesh (EMV). I do not understand Section 5.3 on the proof of (3.10). They want to prove that the relative ...
- 429
1
vote
0
answers
274
views
Direct solution to maximum likelihood computation problem using the derivative of multivariate Gaussian w.r.t. covariance matrix
For an application, I need to compute the maximum loglikelihood of data coming from a $d$-dimensional multivariate Gaussian random variable:
$$
\textbf{x} \sim \mathcal{N}(\boldsymbol{\mu}, \Sigma)
$$
...
- 11
7
votes
1
answer
442
views
Completely bounded norm for unital maps with completely positive sections
Consider a completely bounded unital map $\Phi: \mathbf M_h(\mathbb C) \to \mathbf M_k(\mathbb C)$. Suppose that $\Phi$ has right-inverse $\Psi$ which is completely positive. Is the operator norm of $\...
- 1,444
5
votes
1
answer
534
views
Image of the trace map of ring of integers
Let $L/\mathbb{Q}$ be a finite Galois extension, and let $\mathcal{O}_L$ be the ring of integers of $L$.
We have $tr_{L/\mathbb{Q}}(\mathcal{O}_L)=d\mathbb{Z}$ for some $d\geq 1.$
Fact. $d=1$ if ...
- 1,766
0
votes
1
answer
205
views
Trace of a finite hypercubic tensor
Is the trace of a finite hypercubic tensor defined?
Clearly, for the bidimensional case $n \times n$ the trace is defined as the sum of the elements on the main diagonal:
$$\operatorname {tr} (\...
- 143
4
votes
2
answers
3k
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
$$\max \...
- 41
4
votes
1
answer
266
views
Norm/trace of product inequality involving skew symmetric matrices
I wonder if the following inequality involving skew symmetric matrices is true:
Suppose that $B,C \in \mathbb{R}^{d \times d}$ are skew-symmetric matrices, and $\Sigma \in \mathbb{R}^{d \times d}$ ...
- 51
3
votes
1
answer
772
views
Norm and trace inequalities
If $A$ and $B$ are two positive definite matrices such that $\|A\| \leq \|B\|$ for every unitarily invariant norm $\| \cdot \|$, and $U$ is an $n\times k$ matrix with adjoint $V$ such that $VU = I_k$, ...
- 41
3
votes
1
answer
428
views
Submodularity property of trace of inverse matrix
$\newcommand{\tr}{\operatorname{tr}}$Does submodularity property hold for the trace of a positive-definite hermitian matrix?
I.e., does given any real symmetric positive-definite matrices $X,A,B$
$$
...
- 131
2
votes
0
answers
198
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\...
- 583
11
votes
3
answers
1k
views
Boundedness of the derivative of the trace of an H^1 function
As a research preface, this question is linked to a problem of increasing magnetism in Ginzburg-Landau equations that I have distilled for the purpose of getting to the bottom of this technical matter....
- 882
3
votes
1
answer
752
views
A distributional normal derivative for functions in $H^1(\Omega)$
Let $\Omega$ be a smooth bounded domain with $\partial\Omega = \Gamma$. I have read this.
For all $u \in H^1(\Omega)$ such that $-\Delta u = g \in L^2(\Omega)$ in distribution, we can define the ...
- 266
4
votes
1
answer
204
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: \...
- 1,444
6
votes
1
answer
1k
views
Every self-adjoint trace class operator on $L^2$ has integral kernel
I have asked this question on MSE but did not receive an answer. I thought I could try it here.
Let $T$ be a self-adjoint trace-class operator on $L^2(\mathbb{R})$. Is is true that it can be ...
- 437
6
votes
2
answers
2k
views
Trace of n-th root of unity in cyclotomic extension of p-adic rationals
Let $n\in\mathbb N$ and $p$ be any prime. Denote by $\mathbb Q_p$ the $p$-adic numbers. For a field extension $L/K$ denote by $Tr_{L/K}$ the corresponding trace function.
Let $\zeta_n$ be a primitve $...
- 115
1
vote
1
answer
802
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 $...
- 255
4
votes
1
answer
4k
views
Numerical trace of inverse matrix from Cholesky
This question was somewhat answered here: Fast trace of inverse of a square matrix. However, I feel like there was no complete answer wrt the Cholesky case.
I have the matrix $\Sigma=LL^T$. Is there ...
- 171
3
votes
0
answers
128
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 ...
- 9,330
2
votes
2
answers
241
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 \...
- 721
2
votes
1
answer
184
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 \...
- 2,712
8
votes
1
answer
1k
views
Efficiently compute the trace of a sparse matrix times the inverse of a sparse matrix?
How can I efficiently compute
$\mathrm{trace}(A(B^{-1}))$
where $A$ and $B$ are both sparse symmetric PSD $n \times n$ matrices, both with $O(n)$ non-zero entries? If it helps, the pattern of non-...
- 500
2
votes
1
answer
350
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 ...
- 21
2
votes
1
answer
810
views
On matrices that almost have the same eigenvalues
Let $A$ and $B$ be two $4\times 4$ matrices. Using Newton's identities, one can prove that if
$$\det(A) = \det(B)\quad \text{and}\quad \mathrm{tr}(A^i) = \mathrm{tr}(B^i)$$ for $i=1,2,3$, then $A$ and ...
- 2,154
2
votes
0
answers
48
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 $s$ ...
- 1,766
2
votes
1
answer
109
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 ...
- 1,444
9
votes
0
answers
267
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' ...
- 11.3k
3
votes
1
answer
2k
views
Trace of finitely generated projective module
Let $k$ be a field and $A$ a $k$-algebra with unit. The trace module is
$$
T(A)=A/[A,A],
$$
where $[A,A]$ is the left $A$-module generated by all elements of the form $ab-ba$ for $a,b\in A$. The ...
user1688
1
vote
0
answers
148
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{...
- 283
10
votes
5
answers
990
views
Non-conjugate words with the same trace
Let n>=2, p a large prime, G = SL_n(Z/pZ).
If n=2, there are words that, while not conjugate in the free group, do have identical trace in G. For example, tr(g h^2 g^2 h)= tr(g^2 h^2 g h) for all g, ...
- 20.2k
4
votes
1
answer
2k
views
Derivative of trace of pseudo inverse
Given three matrices $A$ (broad), $B$ and $C$, I'd like to find the derivative of
\begin{align}
f = \textrm{tr} \{BA^+\} + \textrm{tr} \{B(A^+)^TCA^+B^T\}
\end{align}
with respect to $A$, where $A^+...
- 291
12
votes
1
answer
540
views
Categorical interpretation of disjoint cycle notation for tracing permutations
For a natural number $n\in\mathbb{N}$, let $\underline{n}$ denote the finite set $$\underline{n}:=\{1,2,\ldots,n\}.$$
A permutation $\sigma\in Aut(\underline{n})$ can be uniquely written (up to order) ...
- 8,659