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0
votes
2answers
81 views

Convexity of the Frobenius norm of the product of two matrices

I have the following function for two matrices ${\bf A}$ and ${\bf B}$: $f({\bf A}, {\bf B}) = \| {\bf Y - XAB} \|_F^2 = trace\{({\bf Y - XAB)}^T({\bf Y - XAB)}\}$ where matrices ${\bf X}_{n \times ...
10
votes
1answer
326 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) ...
3
votes
1answer
135 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 ...
2
votes
1answer
129 views

Nuclear vs Integral operators on Hilbert spaces

Consider the Hilbert space $L_2=L_2[0,1]$. Is it true that for each nuclear (trace-class) operator on $L_2$ there exists a function $K\in L_1(L_2)$ such that $$Tf = \int\limits_0^1 K(s) f(s) ...
1
vote
0answers
72 views

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 ...
0
votes
1answer
95 views

reference request: trace/lifting operator for $L^{\infty}$ data in bounded $\Omega\subset R^d$

I know the answer to my question must be somewhere in the literature. Probably in [Adams-Fournier], [Dautray-Lions] or something alike, but I don't have access to a library right now so I'll ask ...
0
votes
0answers
88 views

Bound on integral of elliptic theta function

I need to prove that the following bound is true. I thought this might follow from the inversion property of the theta function, as the infinite sum in the integrand is precisely ...
3
votes
2answers
265 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 ...
1
vote
0answers
108 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 ...
2
votes
1answer
139 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 ...
4
votes
1answer
95 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 ...
2
votes
1answer
255 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 ...
4
votes
0answers
203 views

Probability distribution function for singular value sum of Gaussian random matrix

Let $\mathbf{X}$ be an $N \times N$ random matrix with IID Gaussian entries. They can be standard normal, but $N$ is not large: that is $N$ $<$ 6, typically. Call its singular value decomposition ...
1
vote
0answers
117 views

Estimate the diagonal of a Cholesky factor…?

I'm computing several hundred Cholesky factorizations of large, sparse matrices, and I'm really only doing Cholesky factorization because I need to know the diagonal elements of the Cholesky factor L. ...
0
votes
1answer
263 views

solving trace norm equality [closed]

Problem Formulation under what conditions can we solve $\mathrm{trace}(\mathbf{AB})=0$ ? or more specifically, when will $\mathrm{trace}(\mathbf{AB})=0$ implies that $\mathrm{trace}(\mathbf{B})=0$. ...
3
votes
1answer
345 views

Expected value of trace of matrix inverse

Given a $N\times K$ matrix $A$ of full rank with $ K < N $, a diagonal matrix $D$ and knowing that $E[D]=bI_N$, where $E[\cdot]$ is the expected value and $I_N$ is the $N\times N$ identity matrix ...
4
votes
2answers
258 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 ...
3
votes
2answers
248 views

An expression with an alternating trilinear form, written in terms of the determinant and a symmetric bilinear form

I am trying to understand a line from MacLachlan/Reid's The Arithmetic of Hyperbolic 3-Manifolds (it's in 3.4 if you have the book), that seems it should be elementary but I can't seem to find where ...
8
votes
3answers
353 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 ...
1
vote
1answer
49 views

Is the trace of a Lyaponov transform of a semistable matrix always nonpositive?

Let $A$ be a semistable real matrix (i.e. the real parts of all the eigenvalues of $A$ are nonnegative). Let $P$ be a positive definite matrix. Is it always true that $\text{trace}{A^{T}P+PA} ...
5
votes
1answer
307 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} ...
1
vote
1answer
661 views

Trace in an Infinite dimensional space [closed]

How do we define trace of an infinite dimensional space? How one can compute the trace of an infinite dimensional matrix?
28
votes
2answers
4k 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 ...
5
votes
0answers
204 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) ...
6
votes
2answers
1k views

When is an integral transfrom 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 ...
11
votes
1answer
1k 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 ...
4
votes
2answers
391 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 ...
3
votes
1answer
258 views

Bounds on operator 2-norms on partial traces of linearly related operators

Consider an arbitary positive semidefinite operator ρ, acting on ℂA ⊗ ℂB ⊗ ℂC, for A,B,C finite. Also, let P be an orthogonal projector on ...
38
votes
2answers
2k 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 ...
114
votes
18answers
14k views

Geometric Interpretation of Trace

This afternoon I was speaking with some graduate students in the department and we came to the following quandry; Is there a geometric interpretation of the trace of a matrix? This question ...
0
votes
2answers
591 views

Get rid of tr() in SVM kernel trick

I designed a kernel function (to be used within SVM) which has the expression $tr(AB)$ in it. For efficient implementation of this, I was wondering if I could write $tr(AB)$ as an inner product: ...
1
vote
2answers
535 views

Extremum under variations of a traceless matrix

Sorry for my precedent tentative, I was a little hasty: Ok, I think I'd better put the original problem: I have an action of three fields: $A$ which is the spin-connection, $B$ an skew-symmetric ...
8
votes
5answers
695 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, ...