# 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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### The trace map in étale cohomology

Trace has a pretty formal definition in a monoidal closed category in which $[V, V] \cong V^* \otimes V$ as the composition $\mathbb{Tr} : 1 \rightarrow [V, V] \cong V^* \otimes V \rightarrow 1$, ...

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### 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 $...

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### 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)\...

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### 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 ...

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### 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^{...

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### 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 $...

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### 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 ...

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### 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 ...

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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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 $...

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### 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 ...

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### 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 ...

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### 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}\...

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### 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 ...

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### 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}...

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### 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, ...

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### 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\...

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### 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 $...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 $C$, ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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} \...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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} =...

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### 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 ...

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### 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 ...

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### 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}\...

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### 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 ...

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### 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,...

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### 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 ...

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### 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} (\...

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### 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 ...

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### 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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### 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:
$$
\textbf{x} \sim \mathcal{N}(\boldsymbol{\mu}, \Sigma)
$$
...

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### 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 $s$ ...

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### 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 $\...

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### 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\...

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### 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 ...

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### 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 $\Sigma \in \mathbb{R}^{d \times d}$ ...

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### 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(\...