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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Geometric interpretation of trace

This afternoon I was speaking with some graduate students in the department and we came to the following quandary; Is there a geometric interpretation of the trace of a matrix? This question ...
63 votes
2 answers
8k views

Geometric interpretation of characteristic polynomial

The coefficients of lowest and next-highest degree of a linear operator's characteristic polynomial are its determinant and trace. These have well-known geometric interpretations. But what about its ...
Per Vognsen's user avatar
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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 ...
André Henriques's user avatar
30 votes
3 answers
5k views

When is an integral transform trace class?

Given a measure space $(X, \mu)$ and a measurable integral kernel $k : X \times X \rightarrow \mathbb{C}$, the operator $$ K f(\xi) =\int_{X} f(x) k(x,\xi) d \mu(x),$$ the operator $K$ is Hilbert ...
Marc Palm's user avatar
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15 votes
1 answer
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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 ...
Ricky's user avatar
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14 votes
5 answers
4k 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 ...
Ivanov's user avatar
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13 votes
1 answer
650 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 ...
André Henriques's user avatar
12 votes
1 answer
533 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) ...
David Spivak's user avatar
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12 votes
0 answers
443 views

Are Sobolev trace spaces equal from both sides of the boundary?

Let $\Omega\subset\mathbb R^n$ be a bounded open set and $\Omega'$ the complement of its closure. Assume $\partial\Omega=\partial\Omega'$. Are the quotient spaces $W^{1,p}(\Omega)/W^{1,p}_0(\Omega)$ ...
Joonas Ilmavirta's user avatar
11 votes
2 answers
772 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(...
Nikolayevich's user avatar
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....
Daniel Spector's user avatar
11 votes
1 answer
421 views

Best constant for a trace inequality

Having an open, simply connected set $\Omega \subset \Bbb{R}^N$ we may ask what is the best constant $C$ (if it exists) in the inequality $$ \int_{\partial \Omega} u^2 \leq C\int_{\Omega} |\nabla u|^...
Beni Bogosel's user avatar
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10 votes
5 answers
966 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, ...
H A Helfgott's user avatar
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10 votes
2 answers
460 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
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10 votes
3 answers
2k 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
9 votes
0 answers
254 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' ...
Tomasz Kania's user avatar
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8 votes
1 answer
237 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
8 votes
1 answer
433 views

Ends as a "cotrace" operation on profunctors

As mentioned here, there is a trace operation on the monoidal category of profunctors given by taking coends: for any profunctor $F : A\times X \nrightarrow B \times X$, there is a profunctor $Tr^X(F) ...
Noam Zeilberger's user avatar
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-...
Jeff's user avatar
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7 votes
2 answers
611 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
7 votes
1 answer
339 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 ...
Ioannis Zolas's user avatar
7 votes
2 answers
734 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 \...
nikka's user avatar
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7 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 ...
Peter's user avatar
  • 439
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 $...
Peter's user avatar
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6 votes
1 answer
414 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 $\...
Niel de Beaudrap's user avatar
6 votes
2 answers
296 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
6 votes
0 answers
127 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
  • 493
5 votes
2 answers
384 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
5 votes
1 answer
216 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
5 votes
1 answer
370 views

Trace over the zeros with real part 1/2 Only

If RH is not true, we have that Weil's explicit formula still holds: $$ \sum_{\gamma} h(\gamma) = h(i/2)+h(-i/2)-2 \sum_{n=1}^{\infty} \frac{ \Lambda(n)}{ \sqrt n}g(logn)+\frac{1}{2\pi} \int_{-\infty}...
Mathman's user avatar
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5 votes
1 answer
482 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 ...
GreginGre's user avatar
  • 1,661
5 votes
1 answer
852 views

traces of sobolev spaces under additional assumptions

Let $p\in [1,\infty]$, $\Omega$ an open bounded domain with (smooth, if necessary) boundary $\partial \Omega$. Is there a subspace $X\subset L^p(\Omega)$ - a simply describable space, ideally a ...
Delio Mugnolo's user avatar
5 votes
2 answers
715 views

In what sense do the categorical trace and coend count fixed points?

According to the nlab, the categorical trace of a 1-endomorphism $F:C\to C$ in a 2-category is the set hom$(1_C, F)$ of global elements of $F$. If $F$ is a functor in the 2-category Cat, the ...
Mike Stay's user avatar
  • 1,532
5 votes
1 answer
659 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(\...
Balazs Rath's user avatar
5 votes
1 answer
125 views

Trace spaces on convex polyhedra: compatibility conditions at edges

Let $\Omega$ be a convex polyhedron in $\mathbf{R}^3$ with boundary $\partial\Omega$ consisting of $N$ polygons $\{\Gamma_j\}_{j=1}^N$. It's well known that in general $$ H^s(\partial \Omega) \neq \...
trace_space_question's user avatar
5 votes
0 answers
219 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
  • 13.3k
5 votes
0 answers
111 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
0 answers
96 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
5 votes
0 answers
159 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 $...
David E Speyer's user avatar
5 votes
0 answers
871 views

Trace Theorem for $p=\infty$

I am considering the Sobolev space $W^{1,\infty}(\Omega)$ on a bounded Lipschitz domain $\Omega \subseteq \mathbb{R}^2$. I am wondering whether the trace theorem holds in this case with constant one (...
Stefan's user avatar
  • 59
4 votes
2 answers
1k 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 $\...
yarchik's user avatar
  • 482
4 votes
2 answers
218 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{\...
Yair Carmon's user avatar
4 votes
1 answer
259 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
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^+...
Mkl's user avatar
  • 291
4 votes
1 answer
516 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
4 votes
1 answer
196 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: \...
Niel de Beaudrap's user avatar
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 ...
sachinruk's user avatar
  • 171
4 votes
2 answers
299 views

tracial triples

Say that a triple of real numbers $(a,b,c)$ is a realizable triple if there are matrices $A,B\in SL_2(\mathbb{R})$ such that $tr (A)=a$, $tr (B)=b$, and $tr (AB)=c$. Question: what is the shape of the ...
BN2's user avatar
  • 115
4 votes
2 answers
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 \...
Max M's user avatar
  • 41
4 votes
1 answer
193 views

Does trace handle composition in a traced symmetric monoidal category?

Suppose that $(C,\otimes,I)$ is a traced symmetric monoidal category (TSMC) with symmetrizor $\sigma$ and trace $Tr$. Given two morphisms $f\colon A\to B$ and $g\colon B\to C$, I can tensor them to ...
David Spivak's user avatar
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