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

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1answer
27 views

Functions that are “balanced” on the support of a permutation

Let $F = GF(2^n)$. Let $P(x), Q(x) \in F[x]$ be such that $P(x)$ is a permutation, while $Q(x)$ is not a permutation. For $\lambda \in F^*$ define the function $g_\lambda(x) = Tr(\lambda Q(x))$. Let ...
3
votes
2answers
36 views

For any $A, B \in SL(2, F)$, does knowing $\operatorname{tr}A$, $\operatorname{tr}B$, and $\operatorname{tr}AB$ specify $A$ and $B$?

In title, $F$ denotes a field. Does knowing the trace of two matrices and their product specify those two matrices? Up to some equivalence, perhaps?
3
votes
2answers
249 views

If two matrices have the same trace and determinant, do their powers have the same trace?

Let $A,B$ be two $2 \times 2$ matrices over some finite field $\mathbb{F}_q$, such that they have the same trace and determinant. Does this imply that tr $A^k$ = tr $B^k$ for any integer $k$? I've ...
11
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2answers
248 views

Show that there is always a way to achieve det(A) > 0

a) Assume that $(a_1, ..., a_9)$ are different positive numbers. Let us make a 3x3 matrix $A_s$ by placing them arbitrarily into 9 positions available. Show that there is always a way to assemble ...
4
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1answer
60 views

Coordinate-Free Definition of Trace.

$\DeclareMathOperator{\tr}{trace}$ I am reading the wikipedia article on the trace operator. The section titled Coordinate-Free Definition defines the trace as follows. Let $V$ be a finite ...
2
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2answers
28 views

A Proof of a False Result: If $U$ is $T$-invariant, then so is $U^\perp$.

$\newcommand{\ab}[1]{\langle #1\rangle}\DeclareMathOperator{\tr}{trace}\newcommand{\mc}{\mathcal}$ I have a "proof" of the following wrong fact: Let $T$ be a linear operator on a finite ...
0
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1answer
14 views

Problem getting a product of traces out of a single trace in a computation with unitary matrices

I am stuck in a computation and I would appreciate any help. Let $U$ $$U=e^{i\phi_a\sigma_a}$$ where $\phi(x_0,x_1,x_2,x_3)_a$ are just three real functions and $\sigma_a$ are the Pauli matrices. ...
0
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1answer
45 views

How to understand the definition of Killing form?

Define the matrix commutator $\text{ad}_X$ as $$\text{ad}_XY=[X,Y]=XY-YX$$ where $X,Y\in\mathfrak{g}$ and $\mathfrak{g}$ is the Lie algebra associated to Lie group $G$. Then on Lie group $G$, the ...
4
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1answer
23 views

Relation between tracial states on von Neumann algebras and their GNS representations

Let $M$ be a von Neumann algebra acting on a Hilbert space $H$, and let $\tau$ be a faithful tracial state on $M$. What is the relation between the GNS representation of $(M,\tau)$ and the original ...
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0answers
12 views

Sobolev space trace theory on $M \times [0,T]$

Let $M$ be a compact Riemannian manifold without a boundary. I wonder how the trace map $T:H^1(M \times [0,T]) \to H^{\frac 12}(M \times \{0,T\})$ is exactly.. can I split it into two trace maps for ...
1
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1answer
43 views

Generalized partial trace

I am interested in finding a general rule (from the matrix point of view) for calculating the partial trace. Starting from a matrix $$ A = X_1 \otimes X_2 \otimes \cdots \otimes X_n $$ I know how to ...
3
votes
1answer
44 views

Two Identities Involving Trace

Let $x$ by a $p \times 1$ vector and $M$, $Q$ be two $p\times p$ matrix. Then it is claimed that $$ \mbox{trace }\left( M^{-1}xx^TM^{-1} \right) = \left\| M^{-1}x \right\|^2, $$ and $$ \mbox{trace ...
4
votes
1answer
30 views

Show that trace is a unique linear functional [duplicate]

If $W$ is the space of $n \times n$ matrices over a field, and $f$ is a linear functional on $W$ such that $f(AB)=f(BA)$ for each $A,B$ in $W$, and $f(I)=n$, then $f$ is the trace function. I ...
0
votes
1answer
26 views

Derivative of a trace with respect to a scalar

I have 3 given matrices $A,B$ and $C$ and an unknown scalar $\alpha$. I would like to find the derivative $\frac{\partial f(\alpha)}{\partial\alpha}$ of the following function: ...
3
votes
1answer
27 views

Properties of trace-class operators

Let $X$ be a separable Hilbert space (real or complex). Let $A\in\mathcal{L}\left(X\right)$, a bounded linear operator on $X$, and suppose $B\in\mathcal{L}\left(X\right)$, which is of trace-class. ...
0
votes
1answer
28 views

Matrix Multiplication, Trace and Integration

Let $\omega(x)$ be a $p\times 1$ vector-valued function defined on a random variable $X$ with CDF $F$. Now define $$V:=\int \omega(x)[\omega(x)]^T dF(x).$$ Then define $\gamma$ as follows. $$ \gamma ...
0
votes
0answers
14 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 ...
0
votes
0answers
10 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
votes
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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votes
2answers
32 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 ...
1
vote
0answers
42 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 ...
1
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0answers
38 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 ...
1
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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) := ...
0
votes
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 ...
6
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0answers
94 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 ...
2
votes
4answers
80 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) + ...
0
votes
0answers
31 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
70 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 $tr[(AB)^2]\le tr(A^2B^2)$?
0
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0answers
18 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)?
0
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1answer
47 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) ?
0
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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 $Y$?
4
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 $$\Sigma=\begin{pmatrix}\lambda_1& &0\\ &\ddots&\\0&&\lambda_n\end{pmatrix}$$ ...
0
votes
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 ...
5
votes
4answers
110 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 ...
0
votes
0answers
36 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, ...
0
votes
1answer
54 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
votes
1answer
32 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
votes
2answers
69 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
votes
1answer
62 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
27 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
votes
0answers
15 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 ...
1
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0answers
24 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
169 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 ...
0
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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
votes
1answer
31 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
vote
1answer
58 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
64 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
vote
1answer
71 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 ...