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

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Inclusion of commutators on classical pseudodifferential operators

We denote by $Cl^\mu$ the class of classical pseudo-differential operators of order $\mu$. Consider the notation $$[Cl^{a},Cl^{b}]\hookrightarrow [Cl^{a'},Cl^{b'}]$$ which means that a commutator on ...
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Proof of Hopf's Lemma

This is a segment of the proof to "Hopf's Lemma," from page 348 of PDE Evans, 2nd edition. I have a question regarding this, at the bottom of this post. Proof. 1. Assume $c \ge 0$. We may as well ...
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1answer
33 views

Show that the trace class operators on a Hilbert space form an ideal

Let $(H, (\cdot, \cdot))$ be a separable Hilbert space over $\mathbb{L} = \mathbb{R}$ or $\mathbb{C}$. Suppose that $\{\phi_n\}_{n=1}^\infty$ is an orthonormal basis for $H$. Let $\mathcal{B}(H)$ ...
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1answer
83 views

Derivative of determinant of symmetric matrix wrt a scalar

For a given square symmetric invertible matrix $\mathbf{X}$ and scalar $\alpha$ (such that the entries of $\mathbf{X}$ depend on $\alpha$), I would like to use the following well-known expression for ...
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16 views

Traces and linear combinations of idempotents

Let $R$ be a commutative ring and $S$ a subring, and consider the following conditions on an $n$ by $n$ matrix $A$ with entries in $R$: (a) $A$ is an $S$-linear combination of idempotents of ...
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Reference request: result concerning Leray trace

Let $V$ be a vector space (possibly of infinite dimension). For a linear homomorphism $f\colon V\to V$ define $$N(f)=\bigcup_{n\in\mathbb{N}} \operatorname{ker}(\underbrace{f\circ\ldots\circ ...
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1answer
36 views

Is the trace of an idempotent matrix a sum of idempotents?

Let $R$ be a commutative ring, $n$ a positive integer, and $A$ an idempotent $n$ by $n$ matrix with entries in $R$. Is the trace of $A$ necessarily a sum of idempotents of $R$?
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Extending a trace on algebra to a trace of systems of algebras

Suppose, we have a trace $\tau$ on some algebra $\mathcal{A}$, i.e. $$\tau(aA+bB)=a\tau(A)+b\tau(B)\ \forall A,B\in\mathcal{A}, \forall a,b\in\mathbb{C}$$ The question rises, what are then the ...
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60 views

Derivative of the trace of $X^TP^TPX$ with respect to P

$\newcommand{\Tr}{\operatorname{Tr}}$ Consider the following expression: $\Tr(X^TP^TPX)$ where $X$ and $P$ are real matrices. What is the best way to approach the calculation of its derivative ...
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1answer
80 views

Characterize matrices A such that trace(AC)=0 for every matrix C with trace(C)=0

$A$ is an $n\times n$ matrix on the field $F$ such that for every $n\times n$ matrix $C$ with $\operatorname{trace}(C)=0$ we have $\operatorname{trace}(AC)=0$. Can we characterize such matrices $A $? ...
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160 views

Trace of symmetric positive semidefinite matrix when diagonalized (as a bilinear form) in a non-orthogonal basis

Let $\mathbf{S}$ be symmetric positive semidefinite matrix (i.e. one with all eigenvalues real and non-negative). Then there is an orthogonal matrix $\mathbf{U}$ (with its columns forming an ...
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1answer
151 views

solving equation also involving unknown matrix in trace

Given two real $m$ x $k$ matrices $A_1$ and $B_1$ and two $k$ x $k$ real matrices $A_2$ and $B_2$ I want to solve the following equation for $Q$. $Q$ is an orthogonal matrix, i.e. $Q^TQ=I$. ...
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1answer
63 views

Trace of power of stochastic matrix

I would like to know if this statement is true. Having a stochastic matrix (rows sum up to 1), with a positive (non-negative) diagonal, then it holds that $$\text{trace}({W^2})\leq ...
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1answer
43 views

Trace-zero functions in $W^{1,p}$

This is an excerpt of a textbook's proof for a theorem (Trace-zero functions in $W^{1,p}$), from PDE Evans, 2nd edition, page 275. Next let $\zeta \in C^\infty(\mathbb{R}_+)$ satisfy $$\zeta ...
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2answers
79 views

The trace of an operator

My question is derived from A. Deitmar's book: A First Course in Harmonic Analysis (second edition), p22, Exercise 1.17. Let me rewrite it again: Let $k:\mathbb{R}^2 \rightarrow \mathbb{C}$ be smooth ...
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1answer
32 views

Linear Algebra Question concerning the trace of a symmetric positive definite matrix.

The objective is to minimize the diagonal elements of a symmetric positive definite matrix. The expression of this matrix is a little bit nasty and its inverse is much easier to deal with. Can I claim ...
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46 views

Trace Theorem question

From PDE Evans, page 272. My question is towards the bootom of this post. THEOREM 1 (Trace Theorem). Assume $U$ is bounded and $\partial U$ is $C^1$. Then there exists a bounded linear operator ...
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1answer
54 views

If a series converges, does it converge with additional log term multiplied?

If $\sum_{n} |a_n| < \infty$, is it true that $\sum_{n} |a_n\log(a_n)| < \infty$ if $0 \leq a_n \leq 1$? I am trying to see if $A$ is trace class operator, then $A \log(A)$ is also trace class ...
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1answer
62 views

Norm of symmetric positive semidefinite matrices

I have been researching a lot trying to find an answer to my question and didn't find any so I would appreciate it if anyone can help. If we have 2 symmetric, positive semi-definite matrices $A$ and ...
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2answers
96 views

trace of product of a diagonal and a matrix

I would like to know if anything can be said about the trace of a product of two matrices, where one matrix is a diagonal matrix, i.e., $$\text{trace}(DA)=...$$ Are there some bounds in terms of ...
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78 views

Determinant and trace as conjugations?

For real matrices $A$ it holds that $$\det\,\big(e^A\big)=e^{\mathrm{tr}\,A}$$ so we can write $$\mathrm{tr}=(\exp)^{-1}\circ \;\det\;\circ\;(\exp).$$ Is this interpretation of trace as the ...
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3answers
90 views

Prove that $A$ is diagonalizable iff $\mbox{tr} A\neq 0$

Prove that $A$ is diagonalizable if and only if $\mbox{tr} A\neq 0$. $A$ is an $n\times n$ matrix over $\mathbb{C}$, and $\mbox{rk} A=1$. If $p(t)$ is the characteristic polynomial of $A$, I know ...
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28 views

trace function ($2\times2$) with ordered bases as linear transformation

We got trace function as following: $$\operatorname{tr}\begin{pmatrix} a & b\\ c & d\\ \end{pmatrix}=a+d$$ So now have to write down $[\operatorname{tr}]_{S_1,S_2}$, ...
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A question on Rank and trace of a special matrix [closed]

I want to share the following question which was asked in a competitive exam: For a fixed positive integer $n\geq 3$, let $A$ be the $n\times n$ matrix defined by $A=I-\dfrac{1}{n}J$, where $J$ is ...
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3answers
76 views

Prove or disprove that trace of matrix $X$ is zero

I was trying to solve a question from a competitive exam paper. This is a part of that question. Let $I_n$ and $O_n$ be $n\times n$ identity and null matrices respectively.Let $S$ be $2n\times ...
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1answer
52 views

Trace of the exterior power as a determinant

Let $A$ be a matrix. According to Wikipedia, $$tr(\wedge^k A) = \frac{1}{k!} \det \begin{pmatrix} tr (A) & k-1 & 0 & \cdots \\ tr (A^2) & tr (A) & k-2 & \cdots \\ \cdots & ...
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1answer
37 views

Proof that frobenius norm is a norm [duplicate]

It's pretty basic and I'm sure I'm missing something dumb here, but I'd like to know why $||A+B||_F \leq ||A||_F+||B||_F$ The way I understand it, ...
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12 views

Connes trace doubt and operator $ H=xp$

i am trying to understand the paper from page 315 and on http://www.alainconnes.org/docs/bookwebfinal.pdf a) in the form of a sum of primes what does the integral $$ \int _{Q_{p}} ...
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Formula for trace of particular operators

Let $\mathcal{H}$ be the Hilbert space $L^2(\mathbb{R})$. View the Fourier transform as a unitary operator $\mathcal{F} \in B(\mathcal{H})$. For each function $f \in C_0(\mathbb{R})$, let $T(f) \in ...
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Matrix calculus: rules for partial traces

I'm trying to understand a paper and have trouble seeing why the following can be written: $Tr_E\{[ \rho,V] \} = \sigma Tr_E\{\rho_E V\} - Tr_E\{ V \rho_E \} \sigma$, when we know the following ...
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40 views

optimization problem minimizing trace of a matrix with inverse

I am trying to solve the following problem $\min_{T} \operatorname{trace} \left( A(T^T M T + N)^{-1}A^T\right)$, where $T$ is the matrix I am solving for and $A$ is given, $M\succ0$ and $N\succ0$. ...
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trace of a product of similar matrices

I have two known types of matrices, $$L_j= \left[ \begin{matrix} 1 & 1 \\ f_j & -f_j \end{matrix} \right]$$ and $$P_j= \left[ \begin{matrix} e^{i a_j k_j} & 0 \\ 0 & e^{-i a_j k_j} ...
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1answer
29 views

Linear Algebra,Conjugate Transpose

Let $ M_n(\mathbb C) $ be the space of all $ n\times n $ matrices with complex entries. Prove that function $ \langle, \rangle : M_n(\mathbb C) \times M_n(\mathbb C) \to \mathbb C $ defined by $ ...
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1answer
43 views

Matrix-Trace and Conjugate Transpose (Multiple Choice)

I was trying to solve the following problem from a competitive exam paper. Let $A=( a_{ij})$ be a nXn complex matrix and let $A^*$ denote the conjugate transpose of $A$. Then which of the following ...
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1answer
50 views

Show the relationship between the trace and the number of 4-cycles

Let $G$ be a k-regular graph. Show the exact relationship between $tr(A^4)$ and the number of 4-cycles in $G$. I understand how $tr(A^4)$ tells us the total number of closed paths of length 4 in ...
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187 views

Extension and trace operators for Sobolev spaces

Given that $\Omega \subset \mathbb{R}^{n}$ is an open, convex, Lipschitz bounded set. Let $O \subset \Omega$ be open bounded set then consider. $$u_{m} \rightharpoonup^{*} u \text{ in } ...
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1answer
49 views

Maximizing the trace

Say i have the following maximization. $ max_R$ trace $(RZ): R^TR = I_n$ where $R$ is an $n$ x $n$ orthogonal transformational vector. Also, the SVD of $Z = USV^T$. I'm trying to find the optimal ...
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Prove trace inequality $\mathrm{tr}\{ABCBAD-ABCD-ADCB+CD\} \geq 0$

Let $\mathbf{A}$, $\mathbf{B}$, $\mathbf{C}$, $\mathbf{D}$ be four (generally non-commuting) positive semidefinite matrices of same size. I want to show that (or find a counterexample to) $$ ...
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82 views

Cyclic Properties of the trace in Quantum Field Theory

I'm trying to figure out what's going on in this paper here between lines 10 and 11. But I'll give a brief rundown of what's going on. We are trying to compute the trace of the exponential operator ...
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1answer
178 views

Name of an identity for traceless matrices in $\mathbb{R}^3$?

While working on a more compact presentation of a derivation in the context of incompressible fluid flow we tried to simplify things by introducing intermediate steps instead of writing out lengthy ...
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Trace in non-orthogonal basis

In Dirac notation we can define the trace of an operator in Hilbert space $\rho$ as the follows, $Tr(\rho)=\sum\limits_{|s\rangle \in B} \langle s| \rho |s\rangle$ where B is some orthonormal ...
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211 views

Number of restricted integer $2\times2$ matrices with a given trace

Is there a better method than bruteforcing, to find out the number of possible matrices of order $2\times2$ that have trace $N$. The contraints are that all elements in a matrix must be positive ...
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Showing a particular integral operator is trace class

Let $f$ and $P$ be continuous, integrable functions $\mathbb{R} \to \mathbb{C}$ vanishing at $\pm \infty%$. Concisely, $f,P \in C_0(\mathbb{R}) \cap L^1(\mathbb{R})$. Also, assume that $P$ is ...
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1answer
296 views

Trace of tensor product vs Tensor contraction

I have come across various sources that talk about traces of tensors. How does that work? In particular, there seem to be such an equality: $$ \text{Tr}(T_1\otimes ...
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45 views

Proving $\nabla_A tr(ABA^T C) = CAB + C^T A B^T$

The above equation appears without proof on page 9 (equation 3) of Andrew Ng's notes on Machine Learning I have tried various approaches to prove this to no avail. From the notes it seems that it ...
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114 views

Tricky Question on Induction and Characteristic Polynomials

I am to prove via induction that for any $n \times n$ matrix $A$, the characteristic polynomial of $A$ has degree $n$; $(-1)^n$ as the coefficient of the $\lambda ^n$ terms; $(-1)^{n-1}\cdot ...
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2answers
78 views

Trace of $I+A+A^2$

$a=e^{\frac{2\pi i}{5}}$ $$A=\begin{bmatrix}1 & a &a^2 &a^3 & a^4 \\ 0 & a &a^2 &a^3 & a^4\\ 0 & 0 &a^2 &a^3 & a^4\\ 0 & 0 &0 &a^3 & ...
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118 views

Equivalence of Frobenius norm and trace norm

According to [1], [2] and other related publications, the following holds for any matrix $X$: $$\| ...
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31 views

Trace of a product of a recursive series of non-full rank matrices

Let $H$ be a $m\times n$ matrix that is not full rank. Hence $H^T H \neq \mathbb{1}_{m\times m}$. Can we say anything about the diagonal entries, or the trace, of matrix products of the form: $P_0 = ...
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151 views

To Prove $x'Ax=\mathrm{tr}(xAx')=\mathrm{tr}(Axx')=\mathrm{tr}(xx'A)$

To prove, $x'Ax=\mathrm{tr}(xAx')=\mathrm{tr}(Axx')=\mathrm{tr}(xx'A)$ where  $A$ is a square matrix. $x'$ is the transpose of $x$. For each $x,x'$ are column vector, row vector.