**Are you an EPFL student looking for a semester project?**

Work with us on data science and visualisation projects, and deploy your project as an app on top of GraphSearch.

Concept# Trace (linear algebra)

Summary

In linear algebra, the trace of a square matrix A, denoted tr(A), is defined to be the sum of elements on the main diagonal (from the upper left to the lower right) of A. The trace is only defined for a square matrix (n × n).
It can be proven that the trace of a matrix is the sum of its (complex) eigenvalues (counted with multiplicities). It can also be proven that tr(AB) = tr(BA) for any two matrices A and B. This implies that similar matrices have the same trace. As a consequence one can define the trace of a linear operator mapping a finite-dimensional vector space into itself, since all matrices describing such an operator with respect to a basis are similar.
The trace is related to the derivative of the determinant (see Jacobi's formula).
Definition
The trace of an n × n square matrix A is defined as
\operatorname{tr}(\mathbf{A}) = \sum_{i=1}^n a_{ii} = a_{11} + a_{22} + \dots + a_{nn}
where aii denotes the entry on the

Official source

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related publications

Loading

Related people

Loading

Related units

Loading

Related concepts

Loading

Related courses

Loading

Related lectures

Loading

Related units

Related people

No results

No results

Related publications (13)

Loading

Loading

Loading

Related concepts (83)

Matrix (mathematics)

In mathematics, a matrix (plural matrices) is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns, which is used to represent a mathematical object or a pr

Determinant

In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. I

Eigenvalues and eigenvectors

In linear algebra, an eigenvector (ˈaɪgənˌvɛktər) or characteristic vector of a linear transformation is a nonzero vector that changes at most by a constant factor when that linear transformation is

Conformal field theories (CFTs) play a very significant role in modern physics, appearing in such diverse fields as particle physics, condensed matter and statistical physics and in quantum gravity both as the string worldsheet theory and through the AdS/CFT correspondence. In recent years major breakthroughs have been made in solving these CFTs through a method called numerical conformal bootstrap. This method uses consistency conditions on the CFT data in order to find and constrain conformal field theories and obtain precise measurements of physical observables. In this thesis we apply the conformal bootstrap to study among others the O(2)- and the ARP^3- models in 3D.
In the first chapter we extend the conventional scalar numerical conformal bootstrap to a mixed system of correlators involving a scalar field charged under a global U(1) symmetry and the associated conserved spin-1 current J. The inclusion of a conserved spinning operator is an important advance in the numerical bootstrap program. Using numerical bootstrap techniques we obtain bounds on new observables not accessible in the usual scalar bootstrap. Concentrating on the O(2) model we extract rigorous bounds on the three-point function coefficient of two currents and the unique relevant scalar singlet, as well as those of two currents and the stress tensor. Using these results, and comparing with a quantum Monte Carlo simulation of the O(2) model conductivity, we give estimates of the thermal one-point function of the relevant singlet and the stress tensor. We also obtain new bounds on operators in various sectors.
In the second chapter we investigate the existence of a second-order phase transition in the ARP^3 model. This model has a global O(4) symmetry and a discrete Z_2 gauge symmetry. It was shown by a perturbative renormalization group analysis that its Landau-Ginzburg-Wilson effective description does not have any stable fixed point, thus disallowing a second-order phase transition. However, it was also shown that lattice simulations contradict this, finding strong evidence for the existence of a second-order phase transition. In this chapter we apply conformal bootstrap methods to the correlator of four scalars t transforming in the traceless symmetric representation of O(4) in order to investigate the existence of this second order phase transition. We find various features that stand out in the region predicted by the lattice data. Moreover, under reasonable assumptions a candidate island can be isolated. We also apply a mixed t-s bootstrap setup in which this island persists. In addition we study the kink-landscape for all representations appearing in the t times t OPE for general N. Among others, we find a new family of kinks in the upper-bound on the dimension of the first scalar operator in the "Box" and "Hook" representations.

Related courses (66)

L'objectif du cours est d'introduire les notions de base de l'algèbre linéaire et ses applications.

The purpose of the course is to introduce the basic notions of linear algebra and its applications.

The student will acquire the basis for the analysis of static structures and deformation of simple structural elements. The focus is given to problem-solving skills in the context of engineering design.

Generalisation of the two-way plot based on linear traces introduced by Morgenthaler and Tukey.

2001The two-way plot is a well-known tool to visualize the fit of a two-way table. An additive fit can be represented by two sets of parallel traces, whose intersection ordinates equal the fitted values. Based on linear traces, a variety of other two-way plots were investigated recently. In this work we present two-way plots based on general traces. We show that the resulting fits equal the ones using linear traces. Hence, we also study mixed traces: hyperbolic ones to represent the columns and linear ones to represent the rows of a two-way table.

Related lectures (137)