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

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Semi-finite trace: Equivalent definitions

Let $(N,\tau)$ be a semi-finite von Neumann algebra. This means that $\tau$ is a normal, faithful and semi-finite trace. With $\tau$ one associates the following sets: $$ N_\tau^+ := \{x \in N_+ : \...
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1answer
45 views

Matrix Trace Inequality [on hold]

If $\operatorname{Tr}(A) < \operatorname{Tr}(B)$, is it fair to say that $\operatorname{Tr}(AC) < \operatorname{Tr}(BC)$? All of $A$, $B$ and $C$ are positive definite matrices.
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Dependency and Independency relation in Trace monoid.

I was reading the paper on Trace Theory. Author introduces Dependency and Independecy relations as finite, reflexive and symmetric relation and Independecy relations as symmetric and irreflexive ...
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1answer
28 views

equivalent trace-conditions on $C^*$-algebras

Let $A$ be a $C^*$-algebra and $\tau:A\to\mathbb{C}$ linear. Claim: the following conditions are equivalent: $\tau(ab)=\tau(ba)$ for all $a,b\in A$ $\tau(x^*x)=\tau(xx^*)$ for all $x\in A$ $\tau(uau^...
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Minimizing trace with equality constraints

I would like to solve the following trace-minimization under equality constraints optimization problem: $$W^* =\arg\min \operatorname{Tr}[WCW^T] \text{ s.t. } A=B^TW^TWB$$ where $W,C\in\mathbb{R}^{...
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1answer
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Trace form of Frobenius Norm of Matrix approximation

I'm a CS Student and I've implemented the Convex Non-Negative Matrix Factorization (Convex-NMF) Algorithm for a project. Now, for "classic" NMF algorithms, you get an approximation: $$ \mathbf{A} \...
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0answers
41 views

Is the following equation true about trace of a Matrix?

Is this true? $\operatorname{tr}(AB)^k = \operatorname{tr}(A^k B^k)$ If so, how can one provide proof or counter example? I tried it with the following two matrices and it turned out to be true: \...
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1answer
26 views

On the equivalence of two traces

If we are given $$\rm{Trace}\{ G \: a \: a^T\} = \rm{Trace}\{H \: w \: w^T\}$$ where $a$ is $N \times 1$ vector, $G$ is $N \times N$ symmetrical matrix, and $w^T = [a^T \: t^T \: 1]$ and $t$ is ...
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+50

Calculate the trace of $LBB^*$, where $L:H→H$ and $B:=ΦT^{1/2}$ for some $Φ:U_0→H$, an embedding $ι:U_0→V$ and $T:=ιι^*$

Let$^1$ $U$, $V$ and $H$ be $\mathbb R$-Hilbert spaces $Q\in\mathfrak L(U)$ be nonnegative and symmetric $U_0:=Q^{1/2}U$ be equipped with $$\langle u,v\rangle_{U_0}:=\langle Q^{-1/2}u,Q^{-1/2}v\...
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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 \begin{array}{ccc} D&=&\{a,b\}\...
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Can we write a matrix with zero trace as a commutator? [duplicate]

Let $F$ be a field, with $\text{char}(F) = 0$, and $A \in M_{n\times n}(F)$ with $\text{tr}(A) = 0$. Show that there are matrices $B,C \in M_{n\times n}(F)$ that $A = BC - CB$.
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0answers
20 views

Trace of roots of unity has valuation more than 1

Let $F$ be a finite extension of $\mathbb{Q}_p$ (p is prime) and $K/F$ be a unramified extension of degree $\ell\ge 3$, where $\ell$ is a prime and not equal to $ p$. Denote $\mu_K$ be the group of ...
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Triangle inequality Trace norm

when becomes the triangle inequality for the trace norm an equality? I search for it in books and web, but couldn´t find it. Thanks for help!
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2answers
46 views

Given a matrix $A$ with $\operatorname{tr} (A) = 0$, prove that there is a B such that $\forall 1\leq i\leq n :(B^{-1}AB)_{i,i}=0$

I've tried using some matrices $B^{-1}$ that switch the rows, but the $B$ at the end placed the elements back in the diagonal (in different order) so I couldn't find a rule.
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1answer
82 views

$\sum_{k \geq 1} e^{it \sqrt{\lambda_k}}$ - Theory of distribution

An exercise asks to find the wave trace $w(t)=\operatorname{tr} \left(e^{it \sqrt\Delta}\right)=\sum_{k \geq 1} e^{it \sqrt{\lambda_k}}$ as a distribution (or generalized function) of the Laplacian ...
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2answers
67 views

Prove two complex matrices have null trace

Let $A,B \in \mathbb{C}^{2 \times 2} \setminus \{O_2\}$, where $AB=-BA$ and $\det(A+B)=0$. Prove that $\operatorname{tr}(A) = \operatorname{tr}(B) = 0$ (where $\operatorname{tr}$ is the trace). My ...
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1answer
28 views

Clever way to prove $\langle A,X\rangle=x^TAx$ with $X=xx^T$, $A\in S^n$?

How to prove $\langle A,X\rangle=x^TAx$ with $X=xx^T$, $A\in S^n$? (inner product of matrices) $xx^T$ is rank one. The following is one way to prove it: $$\langle A,X\rangle=\text {tr}(AX)$$ ...
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0answers
63 views

unbounded solution, lim inf of trace,

Show that if $\lim \inf_{t\rightarrow \infty} \int_{t_0}^t \operatorname{tr}\left(A(s)\right)ds= \infty $ then the linear first-order system $x'(t)=A(t)x(t)$ where $A \in C\left(I, \mathbb{R}^{n\times ...
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1answer
58 views

Generalized Poincaré Inequality on H1 proof.

let's see if someone can help me with this proof. Let $\Omega\subset\mathbb{R}^n$ be a bounded domain. And let $L^2\left(\Omega\right)$ be the space of equivalence classes of square integrable ...
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0answers
34 views

Evaluate the product $\DeclareMathOperator{tr}{tr}\tr(AB)\tr(CB^{-1})$

Let $A,C$ be given positive semidefinite matrices, $B$ be an arbitrary positive definite matrix. How can I estimate the value of $\tr(AB)\tr(CB^{-1})$ ? Is that true $\tr(AB)\tr(CB^{-1}) \geq \tr(AC)$ ...
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0answers
25 views

Showing Left Side to Right Side.

Let $\mathbf x$ is a $(p\times 1)$ vector, $\mathbf\mu_1$ is a $(p\times 1)$ vector, $\mathbf\mu_2$ is a $(p\times 1)$ vector, and $\Sigma$ is a $(p\times p)$ matrix. Now I have to show $$-\frac{1}{...
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3answers
75 views

$\operatorname{trace}(AB) = 0$ and $\operatorname{rank} (A)=1$. Prove: $ABA=0$

I know that $AB-BA=A \iff$ $A$ is singular. $A$ and $B$ can be complex. Any hints?
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0answers
10 views

Trace of the heat operator $Z(t)=\sum_{m,n=1}^{\infty}e^{-\frac{\alpha_{m,n}^2}{r_0^2}t}$

I know that the spectrum of the disk of radius $r_0$ is $\lambda_{m,n}=\frac{\alpha_{m,n}^2}{r_0^2}$, where $\alpha_{m,n}$ is the n-th root of the Bessel's function of order $m$. I have to find the ...
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1answer
33 views

The property that det(A) = prod of A's eigenvalues, and tr(A) = sum of A's eigenvalues

Do these two properties fail to be true, if A's characteristic polynomial fails to split? If so, then do we usually work in a vector space with the ground field = $\mathbb{C}$, when we want to use ...
2
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1answer
60 views

How can we prove that the space of trace class operators on a Hilbert space $H$ is the closure of $H\otimes H$ with respect to the trace norm?

Let $(H,\langle\;\cdot\;,\;\cdot\;\rangle)$ be a separable Hilbert space over $\mathbb R$ $\mathfrak L^1(H)$ be the space of trace class operators on $H$ and $$\operatorname{tr}L:=\sum_{n\in\mathbb ...
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0answers
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Demonstration involving inequality of traces of product of psd matrix

Let, $ \forall i \in [1, N]: P_i \in \mathbb{R}^{n \times n}, P_i \succ 0, w_i \in \mathbb{R}, \bar{P} = \sum_{i=1}^N w_i P_i$. Then, I want to demonstrate that $ \sum_{i=1}^N w_i \operatorname{tr}\...
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0answers
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An equality in the proof of Proposition 3 of Section 2.7 of Pierre Samuel's Algebraic Theory of Numbers

I am reading Pierre Samuel's Algebraic Theory of Numbers. I get stuck at an equality within the proof of Proposition 3 of Section 2.7. The statement of the proposition is as follows: Proposition 3. ...
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0answers
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Calculate the trace of $T_nL$ where $L\in L(H)$, $T\in L(H,L(H))$ and $T_n:=\langle T,e_n\rangle_H$ for some ONB $(e_n)_n$ of a Hilbert space $H$

Let$^1$ $(H,\langle\;\cdot\;,\;\cdot\;\rangle)$ be a Hilbert space over $\mathbb R$ $(e_n)_{n\in\mathbb N}$ be an orthonormal basis of $H$ $T\in\mathfrak L\left(H,\mathfrak L\left(H\right)\right)$ ...
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1answer
34 views

Is $\langle A,B\rangle =\operatorname{trace}(AB^T)$ an inner product in $\mathbb R^{n\times m}$?

I don't understand why one should take transpose of $\operatorname{tr}(AB^T)$ and why we use the fact that $\operatorname{tr}(M)=\operatorname{tr}(M^T)$ for any $M$ that is a square matrix to solve ...
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1answer
51 views

How can I solve for a , b , c , d?

Let's say I fix a list of two real numbers $\sigma = (\sigma_1, \sigma_2)$, and I want to show that there exists a real, entrywise-nonnegative matrix $A$ with $\sigma$ as its spectrum. How could I ...
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2answers
54 views

With these two equations, how do I show that either a,b,c,d must be negative, if v is not 0?

If I have the equations $$ad-bc = u^2 +v^2$$ $$a+d = 2u$$ and I want $a, b, c, d \ge 0$, then how I can show that this is impossible, if $v \ne 0$? I.e., if $v \ne 0$, then one of $a,b,c,d$ must ...
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1answer
39 views

Traces inequalities for projectors and reflections

Let $V$ - finite dimension space, $A : V \to V$ positive semidefinite operator, $P$ - orthoprojector ($P^2 = P = P^*$) $R$ - reflection ($R^2 = 1$, $R^* = R$). Can we say something nontrivial (any ...
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0answers
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For which $A$ is true: $tr(XAY)=tr(YAX)$

$n \in \mathbb N,\forall X,Y \in \mathbb K^{n \times n},A \in \mathbb K^{n \times n} $ For which A is true: $tr(XAY)=tr(YAX)$ My answer would be if A is the identity matrix, but is there something ...
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2answers
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$n\cdot tr(AB)=tr(A) \cdot tr(B) $ $A$ is a scalar matrix

Let $A\in M^{n\times n}(\mathbb R)$. prove that if for every other $B\in M^{n\times n}(\mathbb R)$: $n\cdot tr(AB)=tr(A) \cdot tr(B) $, $A$ is a scalar matrix.
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1answer
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Prove Or Disprove: $n\cdot tr(AB)=tr(A) \cdot tr(B) $iff $A$ or $B$ is scalar matrix [closed]

Prove/Disprove: $n\cdot tr(AB)=tr(A) \cdot tr(B) $ iff $A$ or $B$ is scalar matrix. A and B are square matrices of size n. So far I managed to prove that one side is right, left to right, that if A, ...
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0answers
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Derivative of the singular values of the product of two matrices.

If $\mathbf{\Sigma}_{\mathbf{A}^H \mathbf{B}}$ is the diagonal matrix of singular values of $\mathbf{A}^H \mathbf{B}$, what is the derivative of $u_j=\operatorname{Tr}\left(\mathbf{\Sigma}_{\mathbf{A}^...
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1answer
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Solving an equation involving the trace of a field

Let $F$ be a finite field of order $q$ where $q=2^{n}$ and fix $l\in F\setminus{0}$ with $Tr(l)=0$. I want to determine the number of $a$ such that $$Tr(la)=Tr(la^{-1})=1,$$ where $Tr$ denotes the ...
5
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1answer
87 views

Prove that trace of a matrix is $0$.

Let $ n\geq 2 $ and $ A,B,C \in M_{n}(\mathbb{C}) $ be three matrices so that $$ A^{2}B+BA^{2}=2ABA $$ and $ C=AB-BA $. Prove that $ \mbox{tr}(C^{k})=0,\forall k\in \mathbb{N}. $ I tried solving it ...
2
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1answer
56 views

How is the $H^{1/2}$ norm of function defined on a subset of the boundary?

Let $\Omega\subset \Omega^d$, $d\in \{2,3\}$, be a bounded $d$-polyhedron with $n$ faces. Denote the faces of $\partial\Omega$ as $\{e_i\}_{i=1}^n$. Let $u\in H^{1/2}(\partial\Omega)$ Taking the ...
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2answers
329 views

A conjecture about traces of projections

Let $M_n$ denote the space of all $n\times n$ complex matrices. Define $\tau:M_n\rightarrow \mathbb{C}$ by $$\tau(X)=\frac{1}{n}\sum_{i=1}^n x_{ii},$$ where of course $X=[x_{ij}]\in M_n$. Recall that ...
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1answer
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Algebraic Number Theory,Marcus, Chapter 2, Question 16

In question 16 of chapter 2 in Marcus Book, I have to show that $\sqrt{3}\not\in\mathbb{Q}(\alpha)$,where $\alpha=\sqrt[4]{2}$ using the trace idea. the proof starts by assuming that $\sqrt{3}=a+b\...
2
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1answer
50 views

Evaluate the mininal eigenvalue of positive definite matrix

Let $\{A_n\}$ be a sequence of positive definite matrices. Denote the minimal eigenvalue of matrix A by $\lambda_{min}(A)$. If $tr(A_n) \to +\infty$ when $n \to +\infty$, then what can we say about $\...
2
votes
1answer
79 views

Proving existence of an element of trace 1

Let $F=\mathbb{F}_{q}$ be a finite field of order $q=2^{n}$ and let $\beta$ be a primitive element of $F$. I would like to prove that if $q>4$, then for each $1\leq i \leq \frac{q-2}{2}$, there ...
2
votes
2answers
54 views

Evaluate product of trace of two matrices

I have a question. Let A be a positive semi-definite matrix, H be a positive definite matrix. Is the following inequality right: $Tr(AH).Tr(AH^{-1}) \geq (Tr(A))^2$? I tried to take some concrete ...
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1answer
31 views

Understanding “trace of map” in the definition of harmonic maps

I have difficulty understanding "trace of map" in the definition of harmonic map. Let $\phi: (M,g)\to (N,h)$ is map between two Riemannian manifolds, the energy density is defined as $$e(\phi)=\...
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1answer
25 views

Trace of Hermitian Positive Semidefinite Matrix

Well, the question I want to ask is as follows. Suppose A and B are Hermitian Positive Semidefinite (PSD) matrices, I wonder if it is possible to prove $Tr(A*(A+B)^{-1})\in (0,1]$ (if it is correct)?...
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0answers
22 views

How to form a matrix using an endomorphism?

Let $f\in \mathrm{End}_\mathbb{R}(\mathbb{R}[x]_{<4})$ be the endomorphism given by $x^2 \dfrac{\mathrm{d}^2}{\mathrm{d}x^2} + \dfrac{\mathrm{d}}{\mathrm{d}x} + 2\mathrm{i}d$. I'm supposed to be ...
5
votes
2answers
127 views

$Tr(A^2)=Tr(A^3)=Tr(A^4)$ then find $Tr(A)$

Let $A$ be a non singular $n\times n$ matrix with all eigenvalues real and $$Tr(A^2)=Tr(A^3)=Tr(A^4).$$Find $Tr(A)$. I considered $2\times 2$ matrix $\begin{bmatrix}a&b\\c&d\end{bmatrix}$ ...
0
votes
2answers
31 views

Prove that $T_p$ is Self Adjoint

Let $P \in M^{\Bbb C}_{n \times n}$ an invertible matrix. Let $T_p$ be a linear transformation $T_p: M^{\Bbb C}_{n \times n} \to M^{\Bbb C}_{n \times n}$ such that: $$T_p(X)=P^{-1}XP$$. for every $X \...