Questions tagged [trace]

Question 1

Consider those $n \times n$ invertible matrices $\mathbf{A}_n$ whose elements are each in $[-1,1]$. What is the lowest possible sum of the terms of the leading diagonal of $(\mathbf{A}_n^\mathsf{T}\...

Question 2

I am trying to implement a loss function from a paper which is: $$\mathscr{L}_{\text{GL}} = \sum_{i, j=1}^n \lVert x_i- x_j \rVert_2^2S_{ij} + \gamma \lVert S\rVert_F^2$$ where $x\in \mathbb{R}^n$ is ...

Question 3

Given a matrix $X \in \mathfrak{u}(N)$, is there a closed formula expressing the traces of powers of $X$ in the symmetric representation, $\mathrm{Sym}$, in terms of traces in the fundamental ...

Question 4

Let $\mathcal{H}$ be a Hilbert space. For $\xi, \eta\in \mathcal{H}$ we define the bounded linear operator $e_{\xi, \eta}$ by $e_{\xi, \eta} (\zeta):= \langle \zeta, \eta \rangle \xi$. I want to prove ...

Question 5

This question arises from looking at two papers: https://arxiv.org/abs/0707.2147 and https://arxiv.org/abs/1609.01254. The second paper restricts itself to the setting of $\mathcal{B}(\mathcal{H})$ ...

Question 6

Suppose that I have two matrices $A$ and $B$ which are both symmetric: $A=A^T, B=B^T$. Moreover, I know how to diagonalize both $A$ and $B$. Now I would like to define $T=A^{1/2}BA^{1/2}$, which is ...

Question 7

For two Hermitian matrices $A$ and $B$, it can be readily shown that $\mbox{tr}(AB)$ is real. However, when we have three Hermitian matrices, is it still true that $\mbox{tr}(ABC)$ is real? I ...

Question 8

Let $A,B \in \mathbb{C}^{n\times n}$ be Hermitian positive‑semidefinite (that is, $A\succeq 0$ and $B\succeq 0$). Claim. Show that $\operatorname{tr}[(A+B)^{2025}] \ge \operatorname{tr}(A^{2025}) + \...

Question 9

For a real positive definite(semidefinite) tensor $\mathcal{A} \in \mathbb{R}^{n \times n \times \dots \times n}$ of order $m$, we can define: The $\textbf{Frobenius norm}$ as $$\|\mathcal{A}\|_F^2 = ...

Question 10

Let $(A, G, \alpha)$ be a $C^{\ast}$-dynamical system, where $G$ is a discrete group acting on $A$ by automorphisms via the action $\alpha.$ Then given a faithful trace $\tau$ on $A$ can we induce a $...

Question 11

It is well known that the Killing form in positive characteristic $p$ is sometime degenerate for $\mathfrak{sl}_n(k)$; for example the case when $n=p=3$. I don't remember exactly where I read that the ...

Question 12

Let $V$ be a finite dimensional $K$ vector space and $T \in L(V)$. Moreover, let $\overline K$ be an algebraic closure of $K$, $\overline V = \overline K \otimes V,$ and $\overline T \in L(\overline V)...

Question 13

I've been given (without a viable proof) the identity $$ \Delta \operatorname{tr} \left( X^k \right) = 2 k (k-1) \, \operatorname{tr} \left(X^{k-2}\right)$$ where $\Delta$ is the Laplacian operator ...

Question 14

I have a limited knowledge of the Laplacian operator and matrix calculus, but I would like to know how to evaluate $$\Delta\operatorname{Tr}(A^2)$$ where $A$ is an $n\times n$ matrix. I’ve been ...

Question 15

$\DeclareMathOperator\PSU{PSU}\DeclareMathOperator\U{U}\DeclareMathOperator\SU{SU}\DeclareMathOperator\O{O}\DeclareMathOperator\SO{SO}\DeclareMathOperator\GL{GL}\DeclareMathOperator\ASL{ASL}\...

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Questions tagged [trace]

Minimising the trace of a matrix $\operatorname{tr}\left((\mathbf{A}^\mathsf{T}\mathbf{A})^{-1}\right)$

Converting item-wise summation of difference between vectors to calculation using trace

Traces of powers in the symmetric representation of $\mathfrak{u}(N)$ in terms of fundamental traces

How to prove $\sum_k e_{\xi_k, \eta_k}$ is trace class?

Defining $s$-inner products on all of $\mathcal{B}(\mathcal{H})$

Efficiently computing the trace of products of diagonalizable matrices

Is the trace of a product of three or more Hermitian matrices real?

Hermitian positive‑semidefinite trace inequality

Relationship between Frobenius norm and trace for positive definite (semidefinite) tensors

Question on existence of a faithful $G$-invariant trace on $A.$

Non-degeneracy of trace form on $\mathfrak{sl}_n(\mathbb F_q)$

Defining trace and determinant of a linear map using Galois theory

On the candidate identity $ \Delta \operatorname{tr} \left( X^k \right) = 2 k (k-1) \, \operatorname{tr} \left(X^{k-2}\right) $

Laplacian in Random Matrix Theory

Prove this sum over a trace orthogonal basis is always proportional to the identity

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