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

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1answer
44 views

Matrix derivative of a special function

I need some help for calculating the matrix derivative of a special function. I have checked Wikipedia and Matrix Cookbook, but could not get the answer or idea. Let us define $f(X)$ as ...
0
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0answers
9 views

On equality of frobenius norms with constraints

For a given matrix A, \begin{equation} \begin{aligned} &&&\| Z \|^2_F = \| A \|^2_F\\ &&& Tr(Z'Z) = Tr(A'A)\\ & \text{subject to} & & Z \succeq 0\\ ...
1
vote
2answers
31 views

Find trace of matrix M^k in optimum way.

I have to find the trace of every matrix $M^1,M^2.....M^k$ in optimum way. One way is to multiply $M$ every time but complexity increases to $k*n^3$. Is there any better approach?
3
votes
1answer
76 views

Show that there are numbers c and d such that F(A) = cTr(A^2) + d(Tr(A))^2,

Suppose F(A) is a quadratic function of a real symmetric matrix, A. This means that there are numbers $f_{ijkl}$ so that F(A) = $\sum_{ijkl}f_{ijkl}a_{ij}a_{kl}$. Suppose that $F(A) = F(QAQ^t)$ for ...
1
vote
1answer
91 views

Trace minimization subject to diagonal constraints

Problem Revisited - Edited for conciseness: We are given two set of data points X [$p \times n$] and Y [$q \times n$]. Let us assume $X = \hat{X} + \tilde{X}$ and $Y = \hat{Y} + \tilde{Y}$ I am ...
0
votes
2answers
50 views

How can I rewrite a trace of a matrix product to a product of matrices?

Is there any way to transform $C=\mathrm{Tr}\left(AX\right)$ to $C=KX$ or $C=KXM$, $K$ and $M$ can be some arbitrary matrices? I mean I want to get rid of the trace operator and keep the matrix $X$.
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0answers
23 views

Trace and Determinant of Field Extension

In algebra, we had a look at the trace and the determinant of a field extension. I am familiar with those concepts in linear algebra and I have seen that finite extensions can be viewed as a finite ...
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1answer
43 views

Proving the following formulae

I need to the following formula: let $g,h\in SL_2\mathbb{R}$, clearly $[g,h]\in SL_2\mathbb{R}$ since its determinant is one. Reading a publication I found the following equality: ...
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2answers
34 views

additive character of a finite field, trace map to middle field

The additive character of a finite field $\mathbb F_q$ is obtained by using the trace function to base field $F_p$. Can we write some of them into a middle field. Using trace function from $\mathbb ...
2
votes
1answer
30 views

Commutation of a partial trace with an operator

Let the partial trace $\mathrm{tr}_B$ be a mapping from an endomorphism End$\left( H_A\otimes H_B \right)$ onto an endomorphism End$\left( H_A \right)$. Then the partial trace is defined as $$ ...
2
votes
0answers
33 views

positiveness of product of matrix

If $A$ is a positive definite matrix, B is not sure but $tr(B)>0$ where $tr$ is trace, will $tr(AB)>0$ ? that is the trace of the product of those two matrices. B is not a diagonal matrix or ...
1
vote
1answer
36 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
37 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
257 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
votes
2answers
271 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 $3 \times 3$ matrix $A_s$ by placing them arbitrarily into $9$ positions available. Show that there is always a way to ...
5
votes
1answer
65 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
votes
2answers
31 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
votes
1answer
15 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
votes
1answer
46 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
votes
1answer
25 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
13 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
vote
1answer
54 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
46 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
33 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
29 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
33 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
30 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
17 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
12 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
2answers
40 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 $$|Tr(TS)|\leq\|T\|\cdot|Tr(S)|?$$
0
votes
2answers
37 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
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0answers
43 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
40 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
28 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
41 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
100 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
85 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
34 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
71 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
21 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
48 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
27 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
76 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
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0answers
12 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
114 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
29 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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votes
0answers
46 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
69 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
39 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
71 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 ...