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

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11 views

Trace of an element in a separable field extension

Let $L=K(\alpha)$ be a finite separable field extension of $K$ of degree $n$ and let $\alpha$ have minimal polynomial $f(X)\in K[X]$ with roots $\alpha=\alpha_1,...,\alpha_n$. Write ...
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0answers
8 views

Thresholding in spectra of partial traces of random symmetric matrices

I found an interesting behavior while looking at partial traces of random matrices. This is something I was studying numerically, and I haven't completely ruled out the possibility of numerical ...
0
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1answer
26 views

Question about trace class operators

Let $\cal{H}$ be a Hilbert space, $T$ a bounded linear operator on $\cal{H}$, $S$ a trace class operator, then can one verify that $$|Tr(TS)|\leq\|T\|\cdot|Tr(S)|?$$
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2answers
29 views

Derivative of a function of trace

Suppose $X$ is a diagonal matrix, $X \in \mathbb{R}^{m \times m}$. Let $f\colon\mathbb{R} \to \mathbb{R}$ be a twice differentiable function. Find the following $$\nabla^2_X f(tr(X))$$ where ...
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0answers
39 views

About the tensor product identity: $A=B \otimes C = B \otimes I + I \otimes C$

I am reading about Chern classes in Nakahara's Geometry, Topology and Physics, and am having trouble understanding the equation $$ A=B \otimes C = B \otimes I + I \otimes C \tag{1}$$ where $A,B,C$ are ...
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0answers
33 views

An identity with determinant and trace of a matrix

How to prove the following identity: $$\det(A)=\frac{1}{d!}\sum_{\sigma\in S_d}\mathrm{sgn}(\sigma)\mathrm{Tr}_{\sigma}(A)$$ where $\mathrm{Tr}_{\sigma}(A)$ is defined as following if $\sigma$ is ...
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0answers
27 views

Help with understanding a proof concerning traces of a Galois extension

Let $K$ be a field, $L$ a galois extension of $K$ and $M$ a galois extension of $K$, with $K \subseteq M \subseteq L$. Define the trace of an element $a \in L$ as follows: $$tr_{L/K}(a) := ...
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2answers
40 views

Trace of a matrix $A$

Suppose we are given a matrix $$A = \begin{pmatrix} x & y \\ -y & x \end{pmatrix} $$ where $x,y \in \mathbb{R}$ and $x^2+y^2=1$. Then is, $\textrm{tr}(A)$ not equal to $0$? If yes, then ...
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0answers
89 views

When does analytic in the operator norm imply analytic in the trace class norm?

Consider $U$ a nice compact region in $\mathbb{C}$ with boundary $\Gamma$. Let $S_1$ b the ideal of trace class operators on a separable complex Hilbert space $H$. We will let $\|\cdot \|$ be the ...
3
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4answers
72 views

The trace identity $\text{tr}((A+B)^2) = \text{tr}(A^2) + \text{tr}(B^2) + 2\text{tr}(AB)$

Prove that $$\text{tr}((A+B)^2) = \text{tr}(A^2) + \text{tr}(B^2) + 2\text{tr}(AB).$$ Else show a counterexample. I've tried using the trace properties such as $$\text{tr}(A+B) = \text{tr}(A) + ...
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0answers
28 views

Trace of a product of 2 matrices

I have the following problem. Let $\textbf{Y} \in \mathbb{R}^{n \times q} \; n>q, \textbf{H} \in \mathbb{R}^{n \times n}$ such that $\textbf{H}$ is idempotent ($\textbf{H}^{2} = \textbf{H}$) and ...
3
votes
1answer
68 views

$A,B$ be Hermitian.Is this true that $tr[(AB)^2]\le tr(A^2B^2)$?

Suppose $A,B \in {M_n}$ be Hermitian.Is this true that $tr[(AB)^2]\le tr(A^2B^2)$?
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0answers
16 views

upper bound for the sum of trace related to product of two matrices?

Given A and B positive definite matrices The inequality $\sqrt{4tr(AB)}$ $\leq$ $tr(A+B)$ is lower bound for tr(A+B) is there another inequality for the upper bound, i.e. ?? ≥ tr(A+B)?
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1answer
46 views

relation between trace of product and sum of matrices?

Given A and B positive definite matrices. Is there an inequality relation between trace(AB) and trace(A+B) ?
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0answers
24 views

Singular values and trace?

Given that $X$ and $Y$ are positive definite matrices, how can I bound the singular values $\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?
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votes
1answer
75 views

Solution to an apparently simple Optimization Problem

I'm stuck at a proof of a property that is stated in a paper. Imagine we have a diagonal matrix $$\Sigma=\begin{pmatrix}\lambda_1& &0\\ &\ddots&\\0&&\lambda_n\end{pmatrix}$$ ...
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0answers
11 views

LFSR - feedback polynomial

I want to describe the recursion S$_{t}$=S$_{t-2}$+S$_{t-3}$ with help of a Trace function in $\mathbb{F}_{2}$. I found the feedback polynomial f(x) = $x^3+x+1$ But how to continue ? How can I find ...
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4answers
102 views

Prove $\det(I + B) = 2(1 + tr(B)).$

Let A be a $3\times 3$ invertible matrix (with real coefficients) and let $B=A^TA^{-1}$. Prove that \begin{equation*} \det(I + B) = 2(1 + tr(B)). \end{equation*} I know that \begin{equation*} ...
0
votes
1answer
27 views

Conditional expectation, pinching

Let $\mathfrak{C}$ be a unital $*$-subalgebra of the full matrix algebra $M_n(\mathbb{C}).$ Let $\mathbb{E}_\mathfrak{C}$ be the orthogonal projection from $M_n(\mathbb{C}),$ endowed with the ...
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0answers
28 views

Optimizing the trace of a matrix product

I have a problem where I have a NxT matrix P (lets just assume full rank for now, where N>>T) and a TxN inclusion matrix S. Each column of S must contain exactly one 1 and the rest 0's i.e. 1_T*S = 1, ...
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1answer
44 views

a multiple choice question related to trace of a matrix.

let P and Q are two invertible matrices . and PQ= -QP . then which of the following is true a) trace(P)=trace(Q)=0 c)trace(P) is not equal to trace(Q) c) none of the above. i can show that ...
2
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1answer
29 views

Complicated trace derivative

Given a symmetric matrix $Y$ and matrices $Z$ and $X$ what is the derivative in $Z$ of the trace $$ \text{tr}( (XX^T-YZZ^TY)^T (XX^T-YZZ^TY) )? $$ I have looked all over for straightforward ways of ...
3
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2answers
67 views

Conceptual approach to the formula $\sum_{i=1}^n \det(v_1, \ldots, Av_i, \ldots, v_n)=\text{tr}(A)\det(v_1, \ldots, v_n)$

I answered this question earlier showing that $$\sum_{i=1}^n \det(v_1, \ldots, Av_i, \ldots, v_n)=\text{tr}(A)\det(v_1, \ldots, v_n),$$ and while I am happy with my answer, I feel like there should ...
2
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1answer
61 views

Closed formula for $\sum_{i=1}^n \det(v_1, \ldots, Av_i, \ldots, v_n)$ [duplicate]

Denote $(v_1, \ldots, v_n)$ the matrix that has columns $v_1,\ldots, v_n\in \mathbb{R}^n$. Let $A\in \mathcal{M}_{n\times n}(\mathbb{R})$. Is there a clever way (without expanding LHS and doing ...
1
vote
3answers
44 views

$\langle A,B\rangle = \operatorname{tr}(B^*A)$

"define the inner product of two matrices $A$ and $B$ in $M_{n\times n}(F)$ by $$\langle A,B \rangle = \operatorname{tr}(B^*A), $$ where the {conjugate transpose} (or {adjoint}) $B^*$ of a matrix $B$ ...
0
votes
1answer
25 views

Trace minimization-Revised

The problem is as follows: $\displaystyle\min_{V}$ trace($V^TH^T\Phi HV$)$\\$ s.t. $V^TV=I_d$ in the case when $H$ is not known. When $H$ is known, the solution is given by the eigenvectors ...
0
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0answers
12 views

Compute the derivatives of an equation

I have an equation which is equal to: $(-c/2)ln(x) + (-c/2)tr(diag(B^TSB)x^{-1})$ Where $c$ is a constant, $tr$ represents the trace, $diag$ represents the diagonal. $B$, $S$ and $x$ are three ...
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0answers
22 views

A trace inquality for the product of symmetric PSD matrices

I'm estimating the expectation of a quadratic form, using two different estimators, and would like to compare the variances. The first is a MC estimator, and the other is the Hutchinson estimator. I ...
4
votes
0answers
162 views

Conditions for Trace Inequality Tr( ( A² - B² ) Z) >= 0

Consider the $M \times M$ complex positive semidefinite matrices ${\bf A}, {\bf B}, {\bf Z}$. We have the relation $\mu_{\text{max}}{\bf I} \succeq {\bf A} \succeq {\bf B} \succ \mu_{\text{min}}{\bf ...
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0answers
30 views

Intermediate traces???

While pursuing the analogies between some branches of mathematics and some fields of linguistics, I have recently come across the idea of intermediate trace, which, in the framework of the study of ...
0
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1answer
29 views

How does $\inf_{c \in \mathbb{R}} \lVert u - c \rVert_{L^2} \le \lVert \nabla u \rVert_{L^2}$ imply this inequality?

Let $M$ be a compact Riemann manifold with boundary. I want to know, given the inequalities $$ \vert T u \vert_{H^{1/2} (\partial M)} \le C \lVert \nabla u \rVert_{L^2(M)} + \lVert u ...
1
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1answer
52 views

$A$ is the set of all $n \times n$ matrices where $\operatorname{tr}(A)=0$, is $A$ a subspace of $M_{nn}$ (where $n\ge2$)?

$\newcommand{\tr}{\operatorname{tr}}$For $A =$ zero matrix, $$W=\{ A \in M_{nn} : \tr(A) = 0 \}$$ I can proof that the set of all n x n matrices A with $\tr(A)=0$ is a subset of $M_{nn} $for$ \ n \geq ...
0
votes
1answer
58 views

Derivative of a trace function

Let $K$ be a Hermitian matrix, and $X$ be a positive one. What is the derivative of the trace function $$ \mbox{ Tr } X|e^{itK} - X|^3$$ with respect to $t$ at $t = 0$ ? There is a nice formula for ...
1
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1answer
64 views

Integration.Matrix.Determinant.Inverse.Trace.

Given $$ I_n=\int_0^1\frac{x^n}{x^{2012}-1}{\rm d}x\text{ and }J_n=\int_0^1\frac{x^n}{x^{2013}+1}{\rm d}x\quad\forall n>2012, n\in\mathbb N$$ If the matrix $$\rm A=[a_{ij}]_{3\times3}\text{ where ...
2
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0answers
45 views

algebraic equation with trace

I have a problem which I don't know how to attack. Actually I am not even sure there is a way to do it. Is it possible to solve an equation of this form $$A²-\frac{tr(A)²}{4}Id_{4\times 4}=B$$ where ...
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1answer
39 views

Does $\lim_{N\rightarrow\infty}\frac{tr(A'A)}{N}=0$ imply $\lim_{N\rightarrow\infty}\frac{tr(A)}{N}=0$?

$A$ is an $N\times N$ matrix with bounded row and column norms. Does $\lim_{N\rightarrow\infty}\frac{tr(A'A)}{N}=0$ imply $\lim_{N\rightarrow\infty}\frac{tr(A)}{N}=0$? I know this is true for ...
0
votes
2answers
118 views

How to prove that tr(A) = tr(B) given that B is similar to A [duplicate]

If A and B are similar, how does one prove that tr(A) = tr(B)
0
votes
0answers
52 views

Trace minimization when some matrix is unknown

The problem is as follows: $\displaystyle\min_{V}$ trace($V^TH^T\Phi HV$)$\\$ s.t. $V^TV=I_d$ in the case when $H$ is not known. When $H$ is known, the solution is given by the eigenvectors ...
1
vote
0answers
43 views

Let $A,B$ be two $3\times 3$ matrices with complex entries, such that $BA^2=A^2B$. Prove that $\det(AB-BA)=0$

Problem: Let $A,B$ be two $3\times 3$ matrices with complex entries, such that $BA^2=A^2B$. Prove that $\det(AB-BA)=0$ attempt: $A(AB-BA)=A^2B-ABA=BA^2-ABA=(BA-AB)A=-(AB-BA)A$ So ...
1
vote
1answer
33 views

Can find the determinant of a matrix A of size $n$ in terms of the traces of $A^m$ [duplicate]

We can find the determinant of a matrix A of size $n$ in terms of the traces of $A^m$, for $m=1,…,n$ ? It's det of a matrix with term are traces, but i saw but i can't remember Expressing the ...
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vote
2answers
37 views

Show the Trace$(B)^{2} \leq$ nTrace$(B^{T}B)$

The following definition is needed for my actual question: For $A, B \in \mathcal{M}_{n \times n}$ define $$ \langle A, B \rangle = \text{Trace}(B^{T}A) = \sum_{j=1}^{n}\sum_{i=1}^{n}b_{ij} \, ...
0
votes
2answers
63 views

If $\operatorname{trace}(Z)=0$, then there exist $X$, $Y$, $|Y|\not=0$ satisfy $Z=XY-YX$ [closed]

Lemma: if $\operatorname{tr}(Z)=0$,then there exist $X$, $Y$, $|Y|\not=0$ satisfy $Z=XY-YX$. It's an old problem but I can't prove it.
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2answers
42 views

Does the trace and determinant uniquely determine the eigenvalues of a 3 by 3 matrix with algebraic multiplicity of 2?

I have a 3 by 3 matrix $M$ whose eigenvalues are $a$, $b$, and $b$. The determinant and trace of $M$ are known from its eigenvalues: $det(M)=ab^2$ and $Tr(M)=2b+a$. I wanted to show that if ...
0
votes
1answer
18 views

Prove that $\nabla_X tr(X^TB)= B $ where $B \in \mathbb{C^{m*n}}$ and $X \in \mathbb{R^{m*n}} $

Prove that $\nabla_X tr(X^TB)= B $ where $B \in \mathbb{C^{m*n}}$ and $X \in \mathbb{R^{m*n}} $ and $\nabla_X$ is the derivative with respect to X. How can I prove the above?
0
votes
2answers
80 views

The trace of a symmetric matrix

If A is an $n\times n$ symmetric matrix with eigenvalues $c_1 \ge ... \ge c_n$, and $U$ is an $ n \times k$ semi-orthogonal matrix, with $ k \le n$, how to prove that $\text{tr}(U^TAU) \le ...
0
votes
0answers
13 views

Quadratic form expressed with trace.

I am attempting to prove the following identity: $(x-a)A^T(x-a)=\text{tr}(Ax_cx_c^T)+n(a-\bar{x})^2 \text{tr}(A)$ where $x_c=(x-\bar{x})$ and the orders of the vectors are $n$. I got as far as: ...
3
votes
1answer
44 views

Is the trace of a matrix a norm?

If the matrix norm of A is defined as $\|A\|=\sum_{i,j}|Aij|$ then how do I determine if the sum of the diagonal elements, i.e., the trace is a valid norm? I am not really sure how to approach this ...
0
votes
1answer
49 views

Trace and Spectral norm of a matrix

Let $A_{n\times n}$ be a matrix. How I can show $$\vert \operatorname{trace} (A) \vert \leqslant n \sqrt{\rho(A^T A)}= n \Vert A \Vert_2$$ and if $A$ is symmetric and positive definite, ...
3
votes
1answer
124 views

Maximize trace of matrix equation given two constraints

Let $\mathbf{Q}$ be a rotation matrix and $\mathbf{A}$ and $\mathbf{B}$ be two real-valued matrices of the same size. I want to maximize the function $$ f(\mathbf{Q})=tr\;\mathbf{QA} \qquad ...
1
vote
1answer
70 views

trace inequalities: linear algebra

If S is any $n \times n$ real, symmetric, invertible matrix and D is any $n \times n$ diagonal matrix such that $0\prec D \prec I$ then does there exist a constant $\gamma$ such that: ...