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Mathematics Stack Exchange

Questions tagged [trace]

For questions about trace, which can concern matrices, operators or functions.

If your question concerns the trace map that maps a Sobolev function to its boundary values, please use [trace-map] instead.

In the context of linear algebra, the trace of a square matrix $M$ is the sum of the diagonal entries.

It's almost the same idea in the case of operators on a separable Hilbert space (with conditions of convergence).

In the context of partial differential equations, when we work with an open set having good conditions, we can define what we call a trace operator.

Use it with the appropriate tags.

1724 questions
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Can you check my trace algebra in (d dimensions)?

I am having some trouble locating a source for the trace identities of the gamma matrices in d dimensions. I have attempted to derive them here, where d need not be integer, please could you tell me if you think this is correct? Any additional…
Clumsy cat
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What is the trace of a tensor $A^2$?

I am going through 'INTRODUCTION TO TENSOR ANALYSIS' by myself, and there is something I quite don't get it. Thank you in advance for the answer. Let $A$ be a tensor. A trace of $A$ is defined as $tr(A)=A:I$ where a double dot product between dyad…
lucy
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Upper and lower bounds on the $\mbox{tr}\left( ABA^{T }\right)$

Suppose $A$ and $B$ are square matrices. I would like to find the trace or upper and lower bounds on the trace of $ABA^{T}$, where the lower diagonal and diagonal elements of $B=(b_{ij})$ are all zeros, for instance, $$B = \begin{bmatrix} 0 & b_{1}…
Charles Chou
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Frobenius norm and trace

Given a real, symmetric and positive-definite matrix G we have: Frobenius norm of G = [trace(GG')]^1/2 G' = transposed matrix of G I need to prove that: Frobenius norm of G = trace[(GG')^1/2] Could someone help me please?
Luciana
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equivalent characterisations of trace class operators

Consider the following definition: Let $\mathcal T$ be the family of bounded operators $T$ on the separable infinite dimensional complex Hilbert space $\mathfrak H$ with the following property: $T\in\mathcal T$ if and only if there exists…
Gherardo
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"Dirac operators have finite-dimensional kernels"

I am currently reading the book Heat Kernels and Dirac Operators. The definition of the index of a Dirac operator uses the fact that the kernel is finite-dimensional. Unfortunately the proof is rather short: Proposition $3.37$. If $D$ is a Dirac…
Filippo
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Sobolev trace inequality and Schur`s lemma

I am trying to prove for $s>\frac{1}{2}$, a test function $f$, $$\left \| f|_{R^{d-1}} \right \|_{H^{s-\frac{1}{2}}(\mathbb{R}^{d-1})}\leq C\left \| f \right \|_{H^{s}(\mathbb{R}^{d})}$$ for some constant $C$ depending only on $s$ and the dimension…
JJW
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Is there a function or equation that can trace the trajectory of a space rocket's motion on the three coordinate axes (x ,y ,z)?

To be honest I'm working on a project at the university which is a simulation of launching a rocket into space The problem here is that this simulation needs to be realistic enough for my programming skills It is difficult to predict the trajectory…
Khaldoun Al Halabi
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relationship between trace and eigenvalues

Let A be a 10-by-10 matrix and rank(A)=1 then we want to show that the trace of A is an eigenvalue of A. I know there is one nonzero row in echelon form of A and 10-1=9 free variable. and also I know that the characteristic polynomial is (X^10)-tr…
zeinab
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Frobenius norm and trace of matrix

Given a real, symmetric and positive-definite matrix G. G' = transposed matrix of G Please, could you tell me which class of matrices that satisfy: Frobenius norm of G = [trace(GG')]^1/2 = trace[(GG')^1/2] Thanks
Luciana
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Trace inequality for positive N semidefinite Hermitian matrices when N> 2

I am trying to prove or disprove the following trace inequality for positive-semidefinite Hermitian matrices $A_1$, $A_2$, $A_3$: $$ Eq. (1) \qquad |Tr( A_1 A_2 A_3 )|^2 \le Tr(A_1A_2) Tr(A_2A_3) Tr(A_1A_3). $$ One can use the scale invariance of…
valery shchesnovich
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Simple way to find the trace or the bound of the trace of $ ABA^T$?

Suppose $A$ and $B$ are square matrices, is there any simple way to find the trace or the bound of the trace of $ABA^T$? Thanks. $B$ has the form of \begin{align*} \begin{pmatrix} 0 & a_{1} & a_{2} & a_{3} \\ 0 & 0 & a_{1} & a_{2} \\ 0 & 0 & 0 &…
Charles Chou
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Need help to understand the derivation of Union of Square of 2 sets

I was reading this article. And got stuck understanding this derivation: Consider the alphabet $\Sigma = \{a,b,c\}$. A possible dependency relation is …
kishoredbn
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Trace of a matrix $A$

Suppose we are given a matrix $$A = \begin{pmatrix} x & y \\ -y & x \end{pmatrix} $$ where $x,y \in \mathbb{R}$ and $x^2+y^2=1$. Then is, $\textrm{tr}(A)$ not equal to $0$? If yes, then how is this possible?
Dwaipayan Gupta
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Intermediate traces???

While pursuing the analogies between some branches of mathematics and some fields of linguistics, I have recently come across the idea of intermediate trace, which, in the framework of the study of language (particularly, of syntax) is supposed to…
Javier Arias
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