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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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Trace map for universal bundle of Grassmannian

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 ...
maxo's user avatar
  • 33
10 votes
3 answers
3k views

Trace inequality for non-reversible Markov chain

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} := ...
Alex Damian's user avatar
3 votes
1 answer
232 views

Hattori-Stallings trace

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(...
Qwert Otto's user avatar
1 vote
0 answers
469 views

How to show that the trace of a regularized Laplacian defined on two sphere with radius $h\geq 1$ is diverging logarithmically?

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}}...
Azam's user avatar
  • 81
2 votes
1 answer
171 views

An inequality related to matrix trace

$$Tr(A|A^T U \Sigma U^T|) \leq Tr(AA^T U \Sigma U^T)$$ 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 ...
Wayne's user avatar
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2 votes
0 answers
49 views

What conclusions can I derive from this family of trace inequalities?

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
1 vote
1 answer
168 views

[M,N]≅ [M,R] ⊗ N for E-infinity modules

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 ...
user avatar
4 votes
0 answers
162 views

Dimensionality reduction preserving cyclic traces

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 ...
Paul Christiano's user avatar
0 votes
1 answer
743 views

Bound on the trace of inverse matrix

Suppose $A$ is a positive semi-definite matrix and we can bound its trace as $l \le tr(A) \le L$. I am wondering if it is possible to find the upper and lower bounds on the trace of $A^{-1}$ based on $...
Amin's user avatar
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5 votes
2 answers
388 views

Trace identity for $2 \times 2$ reflections [closed]

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)\...
Lopez's user avatar
  • 59
0 votes
0 answers
135 views

Relationship between singular values, traces and Hermitian conjugate

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 ...
Piotr Lewandowski's user avatar
1 vote
0 answers
75 views

Mixed moments of traces

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^{...
Angel's user avatar
  • 171
3 votes
1 answer
267 views

Extreme points of the set of all traces

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 $...
user3826143's user avatar
2 votes
1 answer
192 views

Another formula for the Schwinger term — problems with a calculation

$\DeclareMathOperator\Tr{Tr}$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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0 answers
100 views

Necessary and sufficient conditions for $\mbox{trace}(A^{-1/2}e^{-tB} (AB+BA) e^{-tB}A^{-1/2}) \ge 0$ for all $t$

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 ...
dohmatob's user avatar
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3 votes
0 answers
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Different generating sets for conjugation invariants of several matrices

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 ...
Nick's user avatar
  • 213
1 vote
1 answer
128 views

Spectrum invariant under (generalised) transpose as operator on trace class operators

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 ...
Frederik Ravn Klausen's user avatar
-2 votes
1 answer
235 views

Proving 2 matrices have the same trace [closed]

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 ...
Shthephathord23's user avatar
1 vote
0 answers
94 views

Show that for all $\delta$, we have $\|u\|^2_{\Gamma}\le c_\delta(\|u\|^2_{\omega(\delta)}+\|u\|_{\omega(\delta)}\|\nabla u\|_{\omega(\delta)})$

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 ...
Falcon's user avatar
  • 410
6 votes
0 answers
133 views

Hölder inequality inside trace

$\DeclareMathOperator\tr{tr}$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 ...
Seven9's user avatar
  • 515
3 votes
0 answers
259 views

Inequalities involving traces of products of hermitian positive semidefinite matrices

$\DeclareMathOperator{\tr}{tr}$ 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 $...
Malkoun's user avatar
  • 5,031
3 votes
1 answer
155 views

Algorithm to minimize $\operatorname{tr}(PAP^TB)$?

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
6 votes
2 answers
307 views

Inner products on super vector spaces

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 ...
BinAcker's user avatar
  • 767
5 votes
0 answers
221 views

Two Hattori-Stallings trace questions

$\DeclareMathOperator\THH{THH}\DeclareMathOperator\Perf{Perf}\DeclareMathOperator\map{map}\DeclareMathOperator\tr{tr}\DeclareMathOperator\HH{HH}\DeclareMathOperator\fib{fib}\DeclareMathOperator\id{id}\...
Maxime Ramzi's user avatar
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3 votes
0 answers
156 views

Tensor product of operator subalgebras and properties of the trace

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 ...
oliverkn's user avatar
  • 139
3 votes
0 answers
220 views

For which representations of a Lie algebra is the induced trace form basic?

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
  • 546
7 votes
2 answers
623 views

If I multiply the coefficients of a trace-class operator with bounded complex numbers is it still trace class?

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, ...
Frederik Ravn Klausen's user avatar
1 vote
0 answers
92 views

Is it possible to define the trace of a function over a rectifiable set?

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\...
hamath's user avatar
  • 121
0 votes
0 answers
60 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 $...
RKC's user avatar
  • 141
2 votes
1 answer
326 views

Singular value of Hadamard product

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
  • 21
4 votes
1 answer
267 views

Which operators on the trace-class operators extend to operators on Hilbert-Schmidt operators?

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
2 votes
0 answers
198 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 ...
Nemis L.'s user avatar
  • 143
3 votes
1 answer
296 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 $C$, ...
Maxime Ramzi's user avatar
  • 14.1k
3 votes
0 answers
125 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 ...
Jakob's user avatar
  • 1,986
2 votes
1 answer
560 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 ...
C.F.G's user avatar
  • 4,165
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 ...
H A Helfgott's user avatar
  • 19.3k
5 votes
1 answer
217 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} \...
TARS's user avatar
  • 51
10 votes
2 answers
464 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 ...
H A Helfgott's user avatar
  • 19.3k
4 votes
1 answer
538 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 ...
H A Helfgott's user avatar
  • 19.3k
3 votes
1 answer
155 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 ...
H A Helfgott's user avatar
  • 19.3k
2 votes
1 answer
422 views

Does there always exist a matrix satisfying certain tracial conditions

Given odd integers $0<a<b$, I want to know if there exists an $n$ by $n$ real valued square matrix $M$ such that $$ M_{ij} = M_{ji} \quad \forall i,j \in \{1,2\dots n\}$$ $$ \sum_{i=1}^n M_{ij} =...
Zach Hunter's user avatar
  • 3,413
5 votes
0 answers
97 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 ...
MathGeo's user avatar
  • 81
1 vote
0 answers
937 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 ...
CComp's user avatar
  • 119
4 votes
0 answers
409 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}\...
Artemy's user avatar
  • 650
3 votes
0 answers
81 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 ...
Capublanca's user avatar
8 votes
1 answer
240 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,...
Peter Samuelson's user avatar
4 votes
0 answers
310 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 ...
Kim's user avatar
  • 4,134
0 votes
1 answer
194 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} (\...
Luca Cappelletti's user avatar
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 ...
Constantin K's user avatar
3 votes
1 answer
762 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} \...
Per Alexandersson's user avatar