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

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33 views

When is the following trace inequality valid?

I have $A = A^T$ (and can have any real eigenvalue) and $B = B^T \succeq 0$ and want to know if the following holds $$ trace(AB) \leq 0 \iff \lambda_{max} (AB) \leq 0 $$ I know that the matrix $AB$ ...
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1answer
42 views

Show that the trace of A is less than n

Let $A$ be an $n\times n$ matrix with complex entries such that $A^k=I_n$ for some positive integer $k$. Show that the trace of $A$ satisfies $$|tr(A)| \leq n.$$ I have no idea how to approach this ...
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0answers
11 views

Can we write $\text{tr}[Q(x)^*\nabla^2u(x)Q(x)]$ for $u∈C^2(ℝ^d)$ and $Q:ℝ^d→\text{HS}(H,ℝ^d)$ in terms of a differential operator?

Let $H$ be a separable $\mathbb R$-Hilbert space $u\in C^2(\mathbb R^d)$ and $\nabla^2u(x)$ denote the Hessian of $u$ at $x\in\mathbb R^d$ $\operatorname{HS}(H,\mathbb R^d)$ denote the space of ...
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0answers
16 views
+50

$ι:U→V$ is an embedding, $Q:=ιι^*$, $L∈𝓛(ℝ^d)$, $Φ∈\text{HS}(U,ℝ^d)$ $⇒$ $\text{tr}LΦ\sqrt Q(Φ\sqrt Q)^*$ doesn't depend on $ι$

Let$^1$ $U$ and $V$ be separable $\mathbb R$-Hilbert spaces $\iota\in\operatorname{HS}(U,V)$ be a Hilbert-Schmidt embedding $Q:=\iota\iota^\ast$ $u:\mathbb R^d\to\mathbb R$ be twice Fréchet ...
1
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1answer
25 views

Trace of a third order tensor

So, I have learned that the trace of a second order tensor $\textbf{Q}^{(2)}$ is given by: $Tr(\textbf{Q}^{(2)})=Q_{11}+Q_{22}+Q_{33}$ Now, I want to calculate the trace of a third order tensor $\...
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0answers
16 views

Optimize an Trace matrix form

In paper " Generalized Low Rank Approximations of Matrixces the Dimension of matrix are follow: $A_i$ is $r$ x $c$ L is $r$ x $l_1$ R is $c$ x $l2$ $D_i$ is $l_1$ x $l_2$ why it says ...
1
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1answer
36 views

How tight is this trace inequality?

I would like to know how tight the following trace inequalities are for real symmetric $A$ and real symmetric $B \succeq 0$ $$\mbox{trace} (AB) \leq \lambda_{\max} (A) \cdot \mbox{trace} (B) $$ or ...
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1answer
19 views

Distributive property on trace norm

I hope this is not a trivial question, basically, if we have trace norm of $A$ defined as $||A||_\star := \operatorname{trace}\left(\sqrt{A^*A}\right) = \sum\limits_{i=1}^{\min\{m,n\}} \sigma_i$, if $...
3
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1answer
42 views

Lower bounding the trace of $A^2$ using the trace of $A^T A$

$\DeclareMathOperator{\tr}{tr}$For a real, square matrix $A$, I believe that one has a simple upper bound on the (absolute value of the) trace of its square in terms of the trace of its Gramian-type ...
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1answer
23 views

There is no trace on Cuntz algebra

Here is a general explanation why purely infinite $C^*$-algebras admit no tracial states: Non-existence Tracial states. Is my following explanation for non existence of trace on Cuntz algebra $O_n$ (...
-1
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1answer
98 views

Prove that $\operatorname{trace}(A) = 0$ if and only if $A^2 = 0$. [duplicate]

Let $A\in M_{n \times n}$ such that rank of $A$ is $1$. Prove that $\operatorname{trace}(A) = 0$ if and only if $A^2 = 0$.
3
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3answers
90 views

Differentiating $\mbox{tr} (ABA^TC)$ w.r.t. $A$

Why is $\nabla_A \mbox{tr} (ABA^TC) = CAB + C^TAB^T$? Here $A, B, C, D$ are all $n \times n$ matrices. $$\nabla_A f(A) = \left[\begin{matrix} \frac{\partial f}{\partial A_{11}}... \frac{\partial f}{...
2
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1answer
26 views

Confusion on symmetric bilinear form exercise

I'm trying to solve an exercise on the symmetric bilinear form $f(A,B)=\operatorname{tr}A\operatorname{tr}B-n\operatorname{tr}AB$. I have already found that $V^\perp, \!^\perp V$ are the scalar ...
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0answers
35 views

Maximizing the trace of a complex matrix

Let's say I have the following maximization problem: $max_U{tr(AU)}$ where $A\in\mathbb{C}$ and $UU^\dagger=1$ I know that for $A\in\mathbb{R}$ and $UU^T=1$ the solution is: $U=XZ^T$ where $X$ ...
1
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1answer
34 views

Equivalence between trace and Euclidean norm

In the paper "On best approximate solutions of linear matrix equations", there is a very small equivalence I don't know where it comes from. Let $A$ be a matrix (either real or complex), and $\|A\|$ ...
1
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1answer
35 views

Trace Theorem and Neumann boundary.

I've been studying Trace Theorem. From PDE Evans, we have THEOREM 1 (Trace Theorem). Assume $U$ is bounded and $\partial U$ is $C^1$. Then there exists a bounded linear operator $$T : W^{1,p}(U) \...
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0answers
24 views

Matrix Algebra equation under a constraint

I have toiled hard with this problem but I have neither been able to find a solution, not prove that no solution exists. $W_1 \in \mathbb{R}^{m \times k}, W_2 \in \mathbb{R}^{n \times k}, X \in \...
2
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1answer
44 views

lower bound for the trace$(A^\dagger A).$

Let $A$ be $n\times n$ complex squared matrix. I want to find a lower bound for $\mathrm{tr}(A^\dagger A).$ What I could find so far is that if $A$ is Hermitian then $$\mathrm{tr}(A^\dagger A) \geq \...
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3answers
42 views

Show that similar matrices have same trace

If $A$ and $B$ are $n\times n$ matrices of a field $F$, then show that $\text{trace}(AB)=\text{trace}(BA)$. Hence show that similar matrices have the same trace. I've done the first part (proving ...
2
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2answers
46 views

is the trace of inverse of positive, positive definite matrix decreasing?

Let $A, B$ be non-negative, and symmetric positive definite matrices. If $A\le B$, i.e., all the entries of $B-A$ are non-negative, is it true that $\mbox{trace}(A^{-1}) \ge \mbox{trace}(B^{-1})$?
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0answers
44 views

Trace of the product of a Lie algebra and Lie group element

Take $U \in SU(n)$ and $X \in \mathfrak{su}(n)$. What is known about \begin{align} \text{Tr} (UX) \end{align} In particular Are there any useful identities that apply here? When does $\text{Tr} (...
0
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1answer
40 views

A confusion about the definition of the “trace” norm

Given a $n \times m$ real matrix $A$ of rank $r$ one can define its SVD as $A = UD V^T$ with $D$ being a $r \times r$ diagonal matrix and $U^TU = V^TV= I$. Here clearly the diagonal entries of $D$ are ...
1
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1answer
18 views

Semi-finite trace on a von Neumann algebra: Equivalent definitions

Let $(N,\tau)$ be a semi-finite von Neumann algebra. This means that $\tau$ is a normal, faithful and semi-finite trace. Normality means that $\tau(x) = \sup_i \tau(x_i)$ if $x \in N_+$ is the limit ...
0
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1answer
47 views

Matrix Trace Inequality [closed]

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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0answers
5 views

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 ...
1
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1answer
29 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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0answers
37 views

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}^{...
1
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1answer
16 views

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 ...
1
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0answers
72 views

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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0answers
30 views

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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0answers
20 views

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
29 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 prime degree $\ell (\neq p)$. Denote $\mu_K$ be the group of roots of unity in $K.$ Does there exist ...
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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.
2
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1answer
85 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 ...
3
votes
2answers
69 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
30 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 ...
2
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1answer
59 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 ...
2
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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}{...
3
votes
3answers
89 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
34 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 ...
3
votes
1answer
73 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 ...
0
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0answers
21 views

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}\...
6
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0answers
74 views

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. ...
1
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0answers
32 views

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 $H$ $T\in\mathfrak L\left(H,\mathfrak L\left(H\right)\right)$ ...
1
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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 ...