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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Show $\operatorname{tr}(X^T X Y^{T}\Lambda Y\Phi) \geq \lambda_n\operatorname{tr}(X^T X Y^{T} Y\Phi)$

Let $X,Y\in R^{n\times p}$, $\Lambda = \operatorname{diag}(\lambda_1,\lambda_2,\cdots,\lambda_n)$, $\lambda_1\geq\lambda_2\geq\cdots\geq\lambda_n\gt 0$, $\Phi=\operatorname{diag}(\mu_1,\mu_2,\cdots,\...
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Orthogonal matrix $O$ maximising $Tr(OM)$.

One can prove that the set $\{ Tr(OM) /$ $O$ $\in$ $ O_n ( \mathbb R ) $ $\}$ has a maximum for a certain orthogonal matrix $O$, furthermore, the application $f$ : $M_n (\mathbb R)$ $\mapsto$ $\...
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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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Computing trace of matrix

Let A be a n×n matrix such that $$[a_{ij}]_{n×n}=\frac{((-1)^i)(2i^2+1)}{4j^4+1}$$ then what is $$1+ \lim_{n\to {\infty}}\left(tr(A^n)^{1/n}\right)$$ I cannot figure out how to calculate trace of $$A^...
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Is the sum of traces of products maximised by positive operators?

I'm interested in maximising a quantity of the form $\sum_M \operatorname{Tr}( P R_M)$. I'm maximising over both $P$ and $\{ R_M\}$ subject to the constraint that $P$ is a real trace-one matrix and ...
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Prove that $\mathrm{Tr}(B^\mathsf{T}Y^{-1}B)$ is independent of $B$

Given diagonal $A\in\mathbb{R}^{n\times n}$ with all eigenvalues larger than $1$, and minimal polynomial $\alpha(\lambda)$. Matrix is called cyclic if its minimal polynomial is equal to characteristic ...
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$tr(A)=tr(A^2)=tr(A^3) \implies A$ is Nilpotent [closed]

Let $A$ be a square matrix with $\mathrm{tr}(A)=\mathrm{tr}(A^2)=\mathrm{tr}(A^3)$. Prove that $A$ is nilpotent. The tip in the question is to find the matrix $B$ which is similar to $A$ and then $A^...
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Dimension of certain matrix subspace

Let $V:= \mathbb{R}^{n \times n}$, $X \in V$ be given and $$ U_X := \lbrace A \in V: \mathrm{trace}(AX) = 0 \rbrace $$ a subspace. I want to find the dimension of $U_X$. If $X = 0$, then clearly $U_X =...
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Trace of $(A+B)^n$ with B an involutory matrix

Consider two matrices $A$ and $B$ that do not commute, so that the binomial theorem does not apply. However, one of them (say $B$) is an involutory matrix, meaning that $B^2 = I$. I am wondering ...
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Matrix trace differentiation correctness

I need some help in verifying if my derivation in matrix differentiation is correct. $\alpha_{n\times n \times m}$ is a tensor, $F(w)_{n \times m} , Z_{n \times d}$ matrix independent of $\alpha$ , $\...
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Two definitions for the trace of a linear operator

The trace of a linear map $f:V\rightarrow V$, $\dim V=n$ finite, can be defined in two ways. One uses the induced linear map on $\bigwedge^nV^*$ given by \begin{align*} T\mapsto\sum_{i=1}^nT(v_1,\dots,...
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Definition of trace

I have noticed that some literature always define the trace as the sum of the diagonal elements. But sometimes, the trace is defined as $\operatorname{tr}(S) = g^{ab}S_{ab}$. These two definitions do ...
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Description of traces on certain $C^\ast$-algebra

Let $A$ be a unital $C^\ast$-algebra with a tracial state $\tau$ (i.e. $\tau(ab)=\tau(ba)$ for all $a,b\in A$) and let $\Omega$ be the set of characters (i.e. multiplicative, unital functionals) on $A$...
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Minimizing trace of a matrix

Hello everyone, I hope this question is not too silly but I do not have much experience with regression. So I am reading this article about data interpolation and I came across minimization of the ...
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$AB=BA$ from $e^{A+B} = e^A e^B$, given Hermitian matrices

Let $A$ and $B$ be Hermitian matrices. If $AB=BA$, we know that $e^{A+B} = e^A e^B$. In this paper, the author showed that $\text{Tr } e^{A+B} = \text{Tr } e^A e^B$ iff. $AB=BA$. As such, $e^{A+B} = ...
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Trace of tensor product identity

Let $A: V\to V$ and $B: W\to W$ be linear operators on vector spaces $V$ and $W$. I know how to prove $$\operatorname{tr}(A\otimes B) = \operatorname{tr}(A)\operatorname{tr}(B)$$ by appealing to a ...
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Understanding the trace operator

On page 102 of his book Strongly Elliptic Systems and Boundary Integral Equations, McLean defines the trace operator $\gamma:\mathcal{D}(\bar{\Omega})\rightarrow\mathcal{D}(\Gamma)$ as $$\gamma u=u|_\...
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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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the two scale convergence of $(u^{out} - u^{int})$ having different trace on opposite face of $Y$.

I have this geometry (given below) and want to find the two-scale convergence of $[u_{\varepsilon}]$ and $\{u_{\varepsilon}\}$. If anyone has the idea or a reference it would be very helpful. Let $\...
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Operations between rank 4 tensor and matrices - definitions and computations

Let $\mathcal{A}$ be a rank 4 tensor and $X$ be a rank 2 tensor (i.e. a matrix). Define the Frobenius inner product (FIP) of real matrices $$ A : B = \operatorname{tr}(A^T B) $$ A side question: if $A$...
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Does $ A^{1/2}$ is a self-adjoint operator?

I am studying the trace operator, and in the proof of the following theorem, there is a step that I do not understand: Theorem: Let $ H $ be a separable Hilbert space, $\{\phi_n \} _ n$ an ...
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Inner product and adjoint mapping in matrices

Let $\mathcal{A}_1:S^n \to S^n$ be defined by $\mathcal{A}_1(P)=-(A^*P+PA)$. How can I find $\mathcal{A}_1^{adj} (Z)$, where $\mathcal{A}_1^{adj}:S^n \to S^n$?
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Einstein summation convention of 2-forms: $T_{ik}\omega_{kj}$

I am familiar with Einstein summation convention that means if the same index name appears exactly twice, once as an upper index and once as a lower index, then that term is understood to be summed ...
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What happens to the trace if you multiply with orthogonal matrices

Suppose we are given matrices $A$ and $Q$. Furthermore denote by $U$ and $V$ orthogonal matrices. For a matrix $Q'$ the equality $Q'=UQV^T$ holds. Since $U$ and $V$ are orthogonal matrices, the ...
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$\underset{K}{\operatorname{arg min}} [tr(KDK^TA)-tr(K^TA)]$ when $K$ is idempotent and $D$ diagonal?

I am trying to find the minimum $\underset{K}{\operatorname{arg min}} [tr(KDK^TA)-tr(K^TA)]$ where $K$ is an idempotent matrix, $D$ a diagonal matrix and $A$ a positive semidefinite matrix ($K, D, A \...
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Does $\text{Tr}A \leq \text{Tr}B \implies \text{Tr}PA \leq \text{Tr}QB$?

Let $A\in M_n$ and $B\in M_m$ be complex positive semi-definite matrices such that $\text{Tr}A\leq \text{Tr}B$. Let $P\in M_n$ and $Q\in M_m$ be projection matrices with the same nullity, i.e., $\ker(...
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Connection between Trace of $A^TA$ and eigenvalues of $A$

I am visiting a course about linear algebra. Our professor proved the equality between the Frobeniusnorm and the Schatten-2-norm. For this he used the following equality for a real matrix A. $\sqrt{(\...
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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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largest singular value of real symmetric matrix

I am trying to find a proof of the following fact: Let $M$ be a real symmetric matrix, then the largest singular value satisfies: $$ \sigma_1(M) = \lim_{k \to \infty} \left[\text{Trace}(M^{2k})\right]^...
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If A and B are Hermitian Matrices, $\text{tr}(ABAB) \geq 0$.

Can anybody help me with this problem. If $A$ and $B$ are Hermitian Matrices, then $\text{tr}(ABAB) \geq 0$. It is easy to show that the trace is a real number, but I cannot prove that it must be non-...
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trace inequality of positive definite matrices and diagonal matrices

Assume that $A,B$ are $\mathbb{R}^{n\times n}$ positive semi-definite matrices and $C\in\mathbb{R}^{n\times n}$ is a diagonal matrix with positive diagonal entries taking values from $[0,1]$. Is it ...
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the norm of the trace as a linear functional on $\mathbb{C}^{n \times n}$

Let $T:\mathbb{C}^{n \times n} \rightarrow \mathbb{C}$ be a linear functional on the space of all $n \times n$ matrices whose entries are complex numbers. I need to show that the norm of the trace $||...
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Trace-class property for integral operators on $\mathcal{H}=L^2(\mathbb{R}^2)$.

In most literature, $\mathcal{J}_1$ is the set of operators that are trace class, i.e. $\mathcal{J}_1=\{\text{Operator }T\,:\,||T||_1<\infty\}$, where $||\cdot||_1$ denotes the Schatten-1 norm. Let ...
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Lower bound on trace of product of matrices?

I know there exists a Cauchy-Schwarz type upper bound for the trace of a product of matrices $A$ and $B$, but what about a lower bound? Regarding the properties of the matrices $A$ and $B$ I am ...
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derivative trace of matrix inverse square

Consider matrix M = $\sum_{j=1}^n p_jf(x_j)f^T(x_j)$, P,X - scalar vectors, $ f=\left(1,x_1,x_2,x_1x_2,x_1^2,\ x_2^2\right)^T, x1,x2 \in (-1, 1), p_j > 0$. I need find solution for $\frac{\partial ...
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Product of matrices and vanishing trace

It is well known that if $A$ and $B$ are respectively symmetric and skew-symmetric, then $\text{Tr}(AB) = 0$. But is there some kind of analogous result for if one of these is hermitian? What I mean ...
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Minimizing trace of pseudoinverse of a matrix

Given symmetric positive semidefinite diagonal rank-$r$ matrix $R \in \mathbb{C}^{m \times m}$, where $r < m$, and scalar $p \geq 0$, I have the following optimization problem in matrix $X \in \...
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How to relate Frobenius norm and trace of a matrix?

I came across an expression relating the Frobenius Norm with Trace as follows : $$\|UU^T\hat X\|^2_F = tr((UU^T\hat X)^T(UU^T\hat X))$$ Could someone explain briefly the equivalence ? THanks
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generalized green's identity over multiple boundary segments

$$ \newcommand{\integral}[3]{\int\limits_{#1} #2 \, \mathrm{d}#3} \newcommand{\tensor}[1]{\mathbf{\boldsymbol{{#1}}}} \newcommand{\fspace}[1]{\mathcal{#1}} $$ The Green formula can be generalized as ...
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Maximizing tr(AB) under a rotation matrix constraint for B? Related to von Neumann's trace inequality

Let $\mathbf{A}$ and $\mathbf{B}$ be real matrices with $\mathbf{A}\overset{\mathrm{SVD}}{=}\mathbf{U}_A \mathbf{S}_A \mathbf{V}_A^\mathrm{T}$. I want to maximize \begin{align} \max_B \mathrm{trace}(\...
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Vanishing partial trace

Given a square matrix $A$ (could be chosen to be unitary, and possibly also involutive if it helps), then how do we find a matrix $M$ such that $$\text{pt}(MA) = 0?$$ On a pure tensor, the partial ...
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A trace norm upper bound proof for a matrix that is a linear combination of rank-1 matrices

I want to prove an upper bound for the trace norm of any matrix $W$ which is a linear combination of matrices $M_i \in \mathbb{M}$, $\mathbb{M} = \{uv^T: u \in \mathbb{R}^d, v \in \mathbb{R}^k, \|u \| ...
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Trace of $3x^2+6y^2=1$

Let $𝑆$ be the surface defined by the equation $3𝑦^2+6𝑧^2=1$. Determine the range of values of $x$ for which the $x$-traces (cross-sections) are non-empty. That is, find all $𝑎$ such that the ...
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53 views

Commuting Matrices and Trace Problem

Question 1 Let $A$, $B$, $C$ and $D$ be real symmetric $n\times n$ matrices. $A$ commutes with $B$ and $C$. $D$ commutes with $B$ and $C$. Suppose $trace(B)=trace(C)$. Does this imply that $trace(AC)=...
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Conditions for equality for the nearest rank-$k$ matrix

my proof for the best rank approximation, using the Frobenius norm, is as follows: \begin{equation} \begin{alignedat}{1} \vert\vert{\mathbf{A-B}}\vert\vert_F^2 &= <\mathbf{A - B}, \mathbf{...

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