# 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.

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### It seems that minimizing the largest eigenvalue of this symmetric positive definite matrix maximizes its trace. How?

Starting from the matrix \begin{equation} H = \begin{pmatrix} x_1 & y_1 & z_1 & 1 \\ \vdots & \vdots & \vdots & \vdots \\ x_n & y_n & z_n & 1 ...
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### Trace of $W^{k,p}(\Omega)$ space is $W^{k-1/p,p}(\partial \Omega)$

I was reading about abstract trace space of $W^{1,2}(\Omega)$. They have defined it as $$W^{1/2,2}(\partial \Omega)=W^{1,2}(\Omega)/W_0^{1,2}(\Omega).$$ Similary I thought we can define abstract trace ...
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### Maximize $\mathbb{E}_{x}\left\{\|H x\|_{2}^{2}\right\}$ subject to ${trace}\left\{Covariance_{x x}\right\}=\alpha>0 .$

Let $\boldsymbol{H} \in \mathbb{C}^{m \times n}$ be a matrix and x be a zero-mean signal vector $x \in \mathbb{C}^{n \times 1}$ that has a covariance matrix $\boldsymbol{\Sigma}_{x x}$. For a known H,...
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### Cyclic trace of Grassmannian matrices

We know trace of matrix satisfies $$\text{tr}(ABCD)= \text{tr}(DABC)$$ if the matrices are taking values on usually numbers. Assume now the Matrices are Grassmannian valued, or they are fermions. By ...
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### why the trace inequality in periodic homogenization, by Rescaling and summing over the $\varepsilon$- cells carries $\epsilon^{2}$ besides gradient?

I am studying periodic Homogenization and two-scale convergence. Simply we have the trace inequality as $$\|u\|_{L^2(\partial \Omega)} \leq C \| u \|_{H^1(\Omega)}$$ where $\Omega$ is the domain with ...
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### Is the trace of the Hessian matrix of the logistic loss function a convex function?

Consider the logistic loss function $$\ell(x, y,w) = \log \left( 1 + \exp \left(- y w^T x \right) \right)$$ where $x \in \Bbb R^d$ is an input sample and $y \in \{0,1\}$ is its label. We know that ...
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### Coordinate-Free Definition of Trace, revisited.

I have some questions about a coordinate free definition of the trace of linear operators. This questions has been asked before in this forum (see [1,2]), but I haven't found the answers of my ...
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### Orthogonal projections and being trace class

On some Hilbert space $\mathcal{H}$, I have two orthogonal projections $P,Q:\mathcal{H}\rightarrow \mathcal{H}$. In fact, $Q=UPU^*$ for some unitary $U$ (Note that $Q$ is indeed another orthogonal ...
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