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Questions tagged [trace]

For questions about trace, which can concern matrices, operators or functions. If your question concerns the trace map that maps a Sobolev function to its boundary values, please use [trace-map] instead.

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Trace of matrix product involving identity and powers

Suppose we know $\text{Tr}(A)=a$. Is there a closed formula for obtaining \begin{eqnarray} \text{Tr}(A(A-I)^n), \end{eqnarray} for any $n=1,2,...$ and with $I$ being the identity matrix? Such products ...
Zarathustra's user avatar
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1 answer
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Is it true that $tr(APA^T)$ > $tr(AQA^T)$, if $tr(P)$ > $tr(Q)$

Assuming $P$ and $Q$ and positive definite matrix. Is it true that $tr(APA^T)$ > $tr(AQA^T)$, when $tr(P)$ > $tr(Q)$. [EDIT after first answer: not just that, actually $P_{ii} > Q_{ii} $ for ...
zvi's user avatar
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-2 votes
4 answers
287 views

To find the trace and determinant of a matrix $A$ satisfying $A^{2023} + A = \left(\begin{smallmatrix}2 &2& 0\\ 0&2&2\\ 0&0&2\end{smallmatrix}\right)$ [closed]

If $A$ is a $3 \times 3$ matrix such that $$ A^{2023} + A = B\quad \mbox{where matrix}\ B\ \mbox{is given by}\quad B = \left(\begin{smallmatrix}2 &2& 0\\ 0&2&2\\ 0&0&2\end{...
Priyans's user avatar
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Minimize $\mathrm{tr}((FK + G)^TQ(FK + G) + K^TRK)$ over block lower-triangular matrices $K$

I want to solve the minimization problem $$ \inf_{K\in\mathcal{K}}\mathrm{tr}\left(\left((FK + G)^TQ(FK + G) + K^TRK\right)\Sigma\right) $$ where $\mathcal{K}$ is the set of block lower triangular ...
calculus_crusader's user avatar
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1 answer
54 views

Norm of a simple extension

We have the following setup: $K$ a field $L= K( a)$ algebraic field extension and $m_a$ its minimal polynomial. We need to show that for each $x\in K$ we have: $m_a(x) = N_{L/K}(x-a)$. I just plugged ...
user1072285's user avatar
1 vote
0 answers
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Divergent Tail Sums of Approximations of Non-trace Class Compact Operators

I'm working on approximations of compact operators that are not trace class, and I'm looking for ways to provide meaningful approximation error estimates for truncated eigenfunction expansions. I ...
user avatar
3 votes
2 answers
91 views

$f=0$ on $\partial\Omega$ implies $f\in H_0^1(\Omega)$

Let $\Omega\subset\mathbb{R}^n$ be an open set. Let $f\in C(\bar{\Omega})\cap H^1(\Omega)$ with $f=0$ on $\partial\Omega$. Claim: Then $f\in H_0^1(\Omega)$ holds. Since $H_0^1(\Omega)$ is the closure ...
MaxwellDgt's user avatar
2 votes
1 answer
61 views

Finite dimensional Irreps (of algebras) with same traces must be equivalent ('page 136' in Bourbaki)

I look for the reference (or proof) of the following fact which is from appendix (B $27$) of Dixmier's book on $C^*$-algebras. Claim: Let $A$ be an algebra (not necessarily commutative) over a field $...
Charles Ryder's user avatar
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Double Trace of the tensor product of the metric tensor with vector fields.

So I am currently preparing for an exam on General Relativity and while reading the notes I stumbled upon this: $$ tr[tr[g \otimes X \otimes Y]]= g(X,Y) $$ Where $$ g=g_{ij} dx^{i}\otimes dx^{j} $$ is ...
Geotrael's user avatar
1 vote
0 answers
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Trace morphism in deligne/milne:s "tannakian categories".

Is there a connection between the trace morphism (1.7.3 on page 10 in https://www.jmilne.org/math/xnotes/tc2022.pdf) and characters of finite groups? I am also trying to understand why 1.7.4 on the ...
Ben123's user avatar
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3 votes
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An "almost" true inequality for Hermitian matrices

Let $A$ be an $N\times N$ Hermitian matrix. For $p+q$ even, consider the following inequality: $$\frac{1}{N}\sum_{i=1}^N (A^p)_{ii} (A^q)_{ii} \geq \Big(\frac{1}{N}\sum_{i=1}^N (A^p)_{ii} \Big) \Big(\...
WunderNatur's user avatar
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0 answers
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Proof of Conjecture on `Block-Orthogonalisation Always Reduces Trace'

I have a symmetric positive definite matrix, ${\bf D}\in\mathbb{R}^{NK\times NK}$, which is made up of $K$, $N\times N$, blocks: \begin{equation} {\bf D} = \begin{bmatrix}{\bf D_{11}} & {\bf D_{12}...
Will Dorrell's user avatar
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1 answer
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Trace of Operators [closed]

Let $H_1$ and $H_2$ be two Hilbert Spaces. Let A be an bounded linear operator between $H_1$ and $H_2$ such that $AA^*$ is traceclass, where $A^*$ denotes the adjoint Operator. Is it true that we have ...
emma bernd's user avatar
1 vote
3 answers
89 views

Inequality involving matrix trace and diagonalisable matrices

Given two real PSD matrices diagonalisable by orthogonal matrices: $A=UDU^T$ and $B=VEV^T$, prove that $$tr(A+B-2(A^{1/2}BA^{1/2})^{1/2})\geq0.$$ We can rewrite the inequality as $$tr(D)+tr(E)\geq tr((...
John WK's user avatar
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Finding general expression for symmetric trace-free tensors (STFs)

Given a symmetric $n$-tensor $I_{\alpha_1 ... \alpha_n}$, its tracefree version is given by $$Q_{\alpha_1 ... \alpha_n}=\sum_{k=0}^{\left\lfloor{\frac{n}{2}}\right\rfloor} (-1)^k \frac{ \binom{n}{k} \...
Sanjana's user avatar
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trace of a matrix squared:a formula

Can someone explain to me why $$\operatorname{Tr}K^2=\sum_{\alpha,\beta} K_{\alpha\beta}K^{\alpha\beta}$$
user122424's user avatar
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2 votes
2 answers
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Minimizing the trace of matrix using matrix calculus

If I have a function $f(X)$ that returns a square matrix, and $f(X)$ is a convex function with respect to matrix $X$, I sometimes see literature just tell me $\partial f(X)/ \partial X = 0$ will ...
Taylor Fang's user avatar
0 votes
1 answer
66 views

A Trace inequality

For $A,B$ positive self-adjoint matrices do we have $$\operatorname{tr}(AB^2A) \leq \|A\|\|B\|\operatorname{tr}(BA),$$ where $\|\cdot\|$ denotes the operator norm, and $\operatorname{tr}$ denotes the ...
DimSum's user avatar
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1 answer
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Does $x^TAx \approx 1$ imply $||A-xx^T||_2 \approx 0$, where $A$ positive definite and trace 1, $x$ is a unit vector?

Suppose $A$ is a positive definite matrix and $\text{tr}(A)=1$, $x$ is a unit vector. If $x^TAx=1-\delta$ where $\delta>0$ is a small number, can we give a upper bound for $||A-xx^T||_2^2=\text{tr }...
qmww987's user avatar
  • 925
0 votes
1 answer
27 views

Trace of product of positive semi definite matrix and matrix with all negative eigenvalues

Suppose $A \in \mathbb{R}^{n \times n}$ is a Hurwitz matrix, that is, all the eigenvalues of $A$ have strictly negative real parts, and that $S \in \mathbb{R}^{n \times n}$ is a symmetric positive ...
sixtyTonneAngel's user avatar
2 votes
1 answer
43 views

Trace of symmetric part of product of 3 positive semidefinite matrices

Suppose we have 3 symmetric positive semi-definite matrices $A,B,C$. How can one prove or give counter-example for the following statement? $$ \mathrm{trace}(AB(A-C)^2 + (A-C)^2BA) \ge 0 $$ where we ...
sixtyTonneAngel's user avatar
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0 answers
35 views

Is $\text{trace}(X^k)$ convex in $X$

As titled, I want to know whether $\text{trace}(X^k)$ is convex in $X\in \mathbb{R}^{n \times n}$. After looking up the wikipedia, I know that when $X$ is hermitian (for real cases, symmetric) and ...
南洋小學生's user avatar
0 votes
1 answer
59 views

Matrix multiplications with SVD

I'm trying to understand the calculation of $SU = U(\sigma^2 I + D^2)$, which I need to prove with the condition $S(\sigma^2 I + WW^\top)^{-1}W = W$. Let $W \in \mathbb{R}^{d \times m}$ and $W = UDV^\...
Lopsio's user avatar
  • 85
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2 answers
127 views

Trace condition implies matrix is unitary

Question: Let $U_a$ ($a=1,2,\cdots,n^2$) be unitary $n \times n$ matrices, and suppose that there exists an $n \times n$ matrix $M$ (EDIT: w.l.o.g. we can restrict $M$ to be diagonal[2]) such that $$\...
Ruben Verresen's user avatar
1 vote
1 answer
48 views

compact and trace operator [closed]

I am study compact and trace classes of operators in Hilbert space. To clarify, I asked the following question, but have not been able to resolve it yet. Consider $L_p \left( \mathbb{R}^d \right) $ ...
ets_ets's user avatar
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1 answer
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What is the definition of $\nabla(wf) \in (L^2(\Omega))^d \}.$

Let $\Omega \subset \mathbb{R}^d$ be a domain and let $w \in L^1(\Omega)$. Define $$ H_w(\Omega) = \left\{ f \in L^2(\Omega) : \nabla(wf) \in (L^2(\Omega))^d \right\}. $$ Give reasonable conditions ...
Mr. Proof's user avatar
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0 answers
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finding the Kernel of a LT with trace

I have encountered a problem that I was able to solve partially. Given a matrix P which is invertible above a field F, we define $T:M_{n}(F)\rightarrow F$ by the following $T(A)=tr(PA)$. Prove that T ...
perplexed's user avatar
  • 147
6 votes
1 answer
160 views

Relaxation of $\min_{H} \text{tr}(H^T P H)$

Let $P \in \mathbb{R}^{N \times N}$ be a given symmetric matrix. Specially, $P$ has all zero entries on its diagonal, and all its off-diagonal entries are positive. And I want to minimize $$\begin{...
南洋小學生's user avatar
1 vote
2 answers
101 views

If the trace of a matrix equals its rank, is it idempotent?

It is well-known and can easily be proven that if a matrix $A$ is idempotent, then its trace equals its rank: $$ A^2 = A \Rightarrow \mathrm{tr}(A) = \mathrm{rk}(A) $$ Does the inverse also hold? If ...
Joram Soch's user avatar
0 votes
0 answers
60 views

Does this matrix exist? A commonly encountered puzzle

Suppose we have $n\times n$ positive definite matrix $S$ and $n\times n$ positive semi-definite matrix $Y$. Let $R$ be a diagonal matrix of indicators, such that WLOG $RSR$ is a principle submatrix of ...
Matterhorn's user avatar
0 votes
1 answer
49 views

Is this basis for complex matrices necessarily a unitary basis?

Let us consider the vector space of complex $n \times n$ matrices. Let $\{ V_i \}_{i=1,2,\cdots,n^2}$ be a trace-orthogonal basis of matrices, i.e., $$ \forall i,j \in \{1,2,\cdots,n^2\} : \quad \...
Ruben Verresen's user avatar
1 vote
1 answer
52 views

Inequalities on the trace of matrix products

For two $n\times n$ hermitian matrices $A$, $B$, we have the trace inequality $$\text{tr}(AB)\leq\sum_{i=1}^{n}\lambda_i(A)\lambda_i(B)$$ where the $\lambda_i(X)$ are the eigenvalues of X ordered in ...
Rell's user avatar
  • 11
1 vote
1 answer
33 views

Convergence in trace

I am having difficulties with this problem. Given a non-negative compact self-adjoint operator $\gamma$ i.e. $\langle \gamma u, u\rangle \geq 0$ for all $u$. Denote the cut-off function $\chi \in C^\...
Larry Baynes's user avatar
-1 votes
1 answer
44 views

For $g,h\in{\rm SL}(2,q)\setminus\{\pm I\}$ with ${\rm tr}(g)={\rm tr}(h)$, how does $h$ ${\rm SL}(2,q)$-conjugated relate to $g^{\pm 1}?$

This question is less open-ended than you might think. The Question: For $g,h\in{\rm SL}(2,q)\setminus\{\pm I\}$ with ${\rm tr}(g)={\rm tr}(h)$, how does $h$ ${\rm SL}(2,q)$-conjugated relate to $g^{\...
Shaun's user avatar
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-1 votes
1 answer
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For $g\in\operatorname{SL}(2,q)$, do we have $\operatorname{tr}(g)=\operatorname{tr}(g^{-1})?$

The Question: For $g\in\operatorname{SL}(2,q)$ and $q=p^r$ for a prime $p$ with $r\in\Bbb N$, do we have $\operatorname{tr}(g)=\operatorname{tr}(g^{-1})?$ Here $\operatorname{tr}(h)$ is the trace of ...
Shaun's user avatar
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0 votes
1 answer
47 views

Computing partial trace of a given kronecker product matrix with respect to the first component

Suppose I have two matrices, given as: $A= \begin{pmatrix} a & b \\ c & d \end{pmatrix}$, $B= \begin{pmatrix} e & f \\ g & h \end{pmatrix}$ Then,the Kronecker product of $A$ and $B$ ...
Anindita Sarkar's user avatar
-1 votes
1 answer
53 views

how to calculate first order derivative of matrix [closed]

I'm having trouble understanding how to calculate the first order derivative of matrix, and was wondering if anyone could help me. The function with respect to P is f(P)=Tr($XPAP^TPA^TP^TX^T$)=$||XPAP^...
popura's user avatar
  • 19
0 votes
0 answers
28 views

For an $n$ by $n$ symmetric matrix $M$, why does $Tr(M)^2/Tr(M^2)>n-1$ imply that M is positive or negative definite?

Let $M$ be an symmetric square matrix of size $n$. I am trying to prove that $Tr(M)^2/Tr(M^2)>n-1$ is a sufficient condition for proving that $M$ is positive or negative definite. If in addition $...
Alex_Wiskunde's user avatar
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0 answers
58 views

Show that trace class is a banach space with respect to norm $\|\|_{S_1}$

Show that trace class is a banach space with respect to norm $\| \|_{S_1}$ where the norm is infinite sum of eigenvalues of $(T^*T)^\frac{1}{2}$ and T is a compact operator in a separable hilbert ...
voroshilov's user avatar
0 votes
1 answer
55 views

Trace of product of three matrices

I have an expression of the form \begin{equation} i\operatorname{tr}(ABC - B^\dagger AC), \end{equation} where $A$ and $C$ are Hermitian matrices, but $B$ is not. $B-B^\dagger := i D$, where $D$ is ...
igunnarsson's user avatar
5 votes
1 answer
136 views

Trace inequality: $\mathrm{Tr}(|\rho^{1-t}x\rho^t|)\leq \mathrm{Tr}(|\rho^{1-t}y\rho^t|)$

I've been researching some operator spaces and have stumbled upon the following problem. Suppose $x,y\in\mathcal{B}(\mathcal{H})$ (for $\mathcal{H}$ separable) such that $0\leq x\leq y\leq 1$. Further ...
J_P's user avatar
  • 2,148
-1 votes
1 answer
50 views

Prove that $Tr(M |\psi\rangle \langle\phi|)=\langle\phi| M |\psi\rangle$

Question: I am studying alone and I found p.76 of this book that: $Tr(M |\psi\rangle\langle\psi)=\langle\psi| M |\psi\rangle$. I want to be sure that my understanding of the formula is correct by ...
OffHakhol's user avatar
  • 719
1 vote
1 answer
69 views

gradient of $\operatorname{tr}((AX)^2)$ w.r.t

For Hermitian matrices $A$ and $X$, I'm wondering how to calculate $\frac{\partial \operatorname{tr}(AXAX)}{\partial X}$. My attempt: Let $f(X) \mapsto \operatorname{tr}(AXAX)$. Expanding $f(X+H) = \...
user1239110's user avatar
1 vote
2 answers
55 views

What is the nuclear/trace norm of a single block matrix?

Given a matrix $X$ which has the following form: $\begin{bmatrix} 0 & A \\ 0 & 0 \end{bmatrix}$, where $A$ is again some matrix. How does the nuclear/trace norm of $||X||_{\text{tr}}$ relate ...
v.tralala's user avatar
  • 299
1 vote
0 answers
39 views

Polynomial Functions on Matrix Representations of a Clifford Algebra

Hi everyone, this is my first StackExchange post, so all tips on how to improve the question are very welcome! The question I would like to ask comes from mathematical physics, so please also tell me ...
Lutzimilian's user avatar
0 votes
1 answer
68 views

Largest value of $\mathbf x^{\mathrm T} \mathbf V \mathbf x$ at condition $\mathbf x^{\mathrm T} \mathbf x = 1$; here $\mathbf V$ is +ve definite [closed]

Question: Consider positive definite matrix $\mathbf V$ of dimension $p × p$ is given. Determine the largest value of $\mathbf x^{\mathrm T} \mathbf V \mathbf x$ at condition $\mathbf x^{\mathrm T} \...
m1ns - MATH's user avatar
0 votes
0 answers
65 views

Diagonal matrices whose trace equals $1$

In linear agebra, is there a special name to matrices $M$ of the form $$M = \begin{pmatrix} a_{11} & 0 & \cdots & 0 \\ 0 & a_{22} & \cdots & 0 \\ \vdots & \vdots & \...
Mako's user avatar
  • 702
-2 votes
2 answers
40 views

Symmetric and skew-symmetric matrices [closed]

Is the product of a symmetric and skew-symmetric matrix symmetric? does this answer remain accurate if the skew-symmetric matrix is replaced by a non-symmetric matrix? If not, what happens then?
spam spam's user avatar
1 vote
1 answer
118 views

Does similarity of matrices preserve sum of principal minors?

I am new to linear algebra; someone on Quora posted this "shortcut" method to find the characteristic equation of a $3\times 3$ matrix. Though they demonstrated it through an example, here's ...
Nothing special's user avatar
0 votes
0 answers
45 views

Matrix inequality with different dimension and square root

I am trying to prove the matrix inequality which came from the Gelbrich distance. The inequality seems to be correct as I substituted some random values, but not 100% sure with that. The inequality is ...
Minhyuk's user avatar

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