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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how to calculate four order derivative of matrix

I'm having trouble understanding how to calculate the high 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
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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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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
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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
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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 ...
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Find an orthogonal basis for vector space of $2\times2$ real valued matrices

Let $V$ be the vector space of $2\times2$ real valued matrices. And $(A, B) \to \operatorname{tr}(A^\top \cdot B^\top)$ Find an orthogonal basis for $V$. I can't figure out how to make $$ \begin{...
Fregheit Meier's user avatar
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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
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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
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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 ...
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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
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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
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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
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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
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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
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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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Schatten $p$-ideals for Hilbert operators

As I was solving exercise E.3.4.2. in Gert Pedersen's Analysis Now, I got two questions. 1. Let $1 \leq p < \infty$. Pedersen defines Schatten $p$-ideals $\mathcal{B}^p (\mathcal{H})$ as the set of ...
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Computing ${\rm Tr}(ABAB)$ with $B$ diagonal

I want to compute ${\rm Tr}(ABAB)$ where $A$ and $B$ are both symmetric, and $B$ is diagonal. Is there a "simple" way to do this? For example, I know that when $B$ is diagonal, ${\rm Tr}(AB) ...
WazyMaze's user avatar
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Log determinant form for matrix raised to another matrix

$$\DeclareMathOperator{\tr}{\operatorname{tr}} \tr(\log A) = \log \det A$$ because of its well-known identity. Importantly, it is easier for a computer to find the log determinant than to compute the ...
Richie Bendall's user avatar
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Maximum and minimum value of trace of unistochastic matrix

Let $P$ be a unistochastic matrix, that is, there exists a unitary matrix $U$ such that $P_{ij}=|U_{ij}|^2$. What is the maximum and minimum value of $\mathrm{Tr}[P]$? I'm also interested in the ...
ytaguchi's user avatar
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$F_{s;I}=\frac1{2^{n-|I|}}\sum_{supp(t)\subset I^c}tr(\chi_tF_{s;I})\chi_t=\frac1{2^n}\sum_{supp(t)\subset I^c}tr(\chi_s\otimes\chi_tf)\chi_t$

Let $f$ be a unitary operator. $\chi_s$ is the stabilizer operator where $s\in \{0,1,2,3\}^n$ where $\chi_s=\bigotimes\limits_{i=1}^n\sigma^{s_i}$ where $s_i\in \{0,1,2,3\}$. So we can write $f=\sum\...
Soham Chatterjee's user avatar
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Is the scalar field ${\bf A} \mapsto \mbox{Tr} \left( {\bf A} {\bf B} {\bf A}^\top {\bf C} \right)$ convex?

Let ${\bf A}, {\bf B}, {\bf C}$ be $d \times d$ (symmetric) positive semidefinite matrices. Let $\mbox{Tr}$ denote the trace operator. Is the scalar field ${\bf A} \mapsto \mbox{Tr} \left( {\bf A} {\...
sabo's user avatar
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Inequality for the trace of the hat matrix in Ridge regression

I was recently reviewing a research paper and came across an inequality expressed as follows: \begin{align} & \text{tr}\Big[\Big(\frac{1}{np}X^\top X + \rho B\Big)^{-1} \Big(\frac{1}{np}X^\top X\...
LL Tony's user avatar
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Prove that $\text{tr}(B_1^{-1} B_2) \geq \text{tr}((A^\text{T} B_1 A)^{+} A^\text{T} B_2 A)$

Suppose that we have two real and positive definite $n \times n$ matrices $B_1$ and $B_2$ and that $A$ is an arbitrary real $n \times n$ matrix. Running some numerical tests by generating random ...
Andreas's user avatar
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Trace Sign of Matrices from Outer Products of Vectors

I’m investigating the properties of the trace of matrices that are formed by the outer products of vectors. Specifically, I’m interested in the sign of the trace for matrices constructed as the ...
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Expectation of matrix-matrix-matrix product

When calculating the vector-matrix-vector product $$\mathbb{E}\left[x_1^TWx_2\right],$$ where $x_1$ and $x_2$ are $n \times 1$ vectors of random variables and $W$ is a $n \times n$ constant matrix, ...
Genius's user avatar
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Evolving quadratic form with heat equation on sphere (to arrive at Trace)

In this answer to MO question "Geometric interpretation of Trace" (the 9th highest upvoted question on the site!), the following interpretation of the trace is given: $$\operatorname{Tr}(A) ...
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Orthogonal Projection onto the Set of Square Matrices with a Unit Trace

The problem is given by: $$ \begin{align*} \arg \min_{\boldsymbol{x}} \quad & \frac{1}{2} {\left\| \boldsymbol{X} - \boldsymbol{B} \right\|}_{F}^{2}, \; \boldsymbol{B} \in \mathbb{R}^{n \times n} \...
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$Tr(A^k) = Tr(B^k)$

Let $A,B \in \mathcal{M}_n(\mathbf{C})$. Let us consider the following assertions : (i) $\forall k \in [[ 1,n ]]$, $\mathrm{Tr}(A^k) = \mathrm{Tr}(B^k)$ (ii) $\forall k \in \mathbf{N}$, $\mathrm{Tr}(A^...
Eric 's user avatar
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The covariance matrix $\mathbf{C}$: Why does $\ln \text{det} \mathbf{C}=\text{Tr}\ln \mathbf{C}$ hold? [duplicate]

Prof. Max Tegmark first introduced the Fisher information matrix into cosmology in his paper titled Karhunen-Loeve eigenvalue problems in cosmology: How should we tackle large data sets? As I read the ...
Stephen Wong's user avatar
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How to calculate the square root of a special matrix

I have met a question to calculate the square root of a matrix in $\mathbb{R}^{s+1}$. $$A = \left(\begin{array}{cc} \sigma_{11}^{2} & \sigma_{11}^{1/2}\Sigma_{1S}\Sigma_{SS}^{1/2}\\ \sigma_{11}^{1/...
T. Wang's user avatar
2 votes
2 answers
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$Tr[(\vec{n_{1}}\cdot\vec{\sigma})(\vec{n_{2}}\cdot\vec{\sigma})] = \sum_{k,l}(n_1)_{k}(n_{2})_l2\delta_{kl}$

Suppose $\vec{n_{1}} = \begin{bmatrix} n_{1x} & n_{1y} & n_{1z}\\ \end{bmatrix}$ $\vec{n_{2}} = \begin{bmatrix} n_{2x} & n_{2y} & n_{2z}\\ \end{bmatrix}$ $\vec{\sigma} = \begin{...
Mathematicing's user avatar
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Trace-preserving map between stochastic matrices

Let $A$ be a stochastic matrix living in a tensor product space $W=V \otimes V \otimes \cdots \otimes V$ of dimension $d^N$, where $dim(V)=d$. In particular, $A$ is a left stochastic matrix, a real ...
Philap's user avatar
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Questions regarding the construction of nilpotent matrices in light of certain characteristics

I have some questions regarding nilpotent matrices. I know that the trace of an $n\ x\ n$ nilpotent matrix must be zero and that the rank of that matrix must be less than $n$. Thus, the determinant ...
charlesFL's user avatar
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1 answer
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Trace Inequality: prove that for arbitrary square matrix A and B: tr((A+B)(A+B)^T) \leq 2 * tr(AA^T+BB^T)

prove that for arbitrary square matrix A and B: $$ tr((A+B)(A+B)^T) \leq 2 \cdot tr(AA^T+BB^T) $$ Here is what i did so far: left side: $$ tr((A+B)(A+B)^T) = tr(AA^T)+tr(AB^T) +tr(BA^T)+tr(BB^T) $$ ...
mads grønbeck's user avatar
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What is computationally the fastest way to calculate $\mathrm{Tr}(A^n)$ and $\mathrm{Tr}(A^{n-1}B)$?

Let $A \in \mathbb{R}^{N\times N}$ be a large symmetric matrix that has at most $\frac{1}{8}$ of its elements non-zero. We have an equation that involves a term $\mathrm{Tr}(A^n)$, that is, trace of ...
Marabellum's user avatar
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1 answer
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Solve for matrix trace equation

How to solve for $\mathbf X \in \mathbb R^{m \times n}$ given: $$ \operatorname{tr}(\mathbf P\mathbf X) = q\,, $$ where $\operatorname{tr}$ is matrix trace, with $\mathbf P \in \mathbb R^{n \times m}$ ...
Kevin's user avatar
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Range of a product of traces

What is the range of $A = tr(q[r, h]) tr(p[r,h])$ where $[m,n]=mn-nm$ is the commutator of $m$ and $n$, knowing that all matrices are hermitian $d \times d$. $ q,p,r $ are trace one, PSD and ...
user18722294's user avatar
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1 answer
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How to calculate the trace of a semigroup?

Let $A$ denotes a self-adjoint operator such that $Tr(A)< \infty$. Let $t\geq 0$. How to show the following \begin{equation} \int_{0}^{\infty}Tr(e^{-tA})< \infty\,. \end{equation} Some facts \...
Aban's user avatar
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Trace of Hermitian involutionary matrix

let $V$ be a Hermitian matrix s.t. $V^2 = I$. Because it's Hermitian, we know that the trace is real. But as it is not necessarily PSD, what does it being involutionary tell us about the trace? I am ...
user18722294's user avatar
2 votes
3 answers
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Let A be a complex $n \times n$ matrix with rank(A) = 1. Why is the minimal polynomial $x(x-Tr(A))$? [duplicate]

We know $rank(A) = 1$ so I have $n-1$ eigenvalues which are $0$ So my characteristic polynomial is $(x-0)^{n-1}(x-a) = x^{n-1}(x-a)$ with $a$ the last eigenvalue to determine. Now, I found on some ...
ReaperSala's user avatar
1 vote
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Trace upper bound with commutator

Let $\rho_0,\rho_1, \rho_2$ be Positive Semidefinite, Hermitian, and trace one complex matrices with $\rho_i^2 = \rho_i$. Let $V$ be Hermitian and $V^{2} = I $. I search for an upper bound of $$|tr(\...
keyla's user avatar
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4 votes
1 answer
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Can $\operatorname{Tr}[A^{-1} BAB^\top]$ be shown to be always positive if $A$ is real and positive definite? [closed]

Let $A$ be a real symmetric positive definite matrix and $B$ is a real matrix with all eigenvalues zero. Can we prove or disprove that $\operatorname{Tr}[A^{-1} BAB^\top]$ is a positive number?
Mike's user avatar
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7 votes
2 answers
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Let $S$ be a finite set of real numbers, and let $T$ be the set of all $n\times n$ matrices having entries in $S$.

Let $S$ be a finite set of real numbers, and let $T$ be the set of all $n\times n$ matrices having entries in $S$. Prove that $$\sum\limits_{A\in T}\mbox{trace}(A^2)=\sum\limits_{A\in T}(\mbox{trace}(...
urt43as's user avatar
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2 answers
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Show $\operatorname{tr}(P) = 1$ where $P$ is an orthogonal projector onto the one-dimensional space spanned by vector b [duplicate]

I am struggle-bussing over this proof right now. This is what I have so far: since $ P $ is a projection, $ P^2 = P$, and the $\operatorname{dim}U = 1$, so by a property, basis $B = b \in \mathbb{R}^{...
CodedRoses's user avatar
1 vote
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Trace of power of Laplacian matrix of a graph

It is well known that, for symmetric matrices, the trace the product of three matrices $A$, $B$, $C$ is the same, independent of the order of $A$, $B$, $C$, that is, $$\operatorname{tr}(ABC) = \...
ChibiPeew's user avatar
6 votes
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300 views

Limit of a particular trace norm.

I have the following problem. Let $\mathbf{\hat{\rho}}(t)$ and $\mathbf{\hat{\sigma}}(t)$ be two trace class positive operators acting on a Hilbert space of infinite dimension for all $t > 0$. More ...
Hldngpk's user avatar
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Upper bound on the trace of the product of real PSD matrices involving the trace of one of them

I am looking at the negative log-likelihood of a multivariate Gaussian $X \sim \mathcal{N}(\mu, \Sigma)$: $$const + (n/2)\log|\Sigma| + (1/2)Tr(\Sigma^{-1}S),$$ where $S = (1/n)\sum(X_i-\mu)(X_i-\mu)'$...
Giora Simchoni's user avatar
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Simplify $\text{Tr}[(A-I)\Sigma_X(A-I)^\top+\Sigma_{\xi}]$

I encounter a problem as below: $$\mathcal{L}(A,\Sigma_{\xi})=\text{Tr}[(A-I)\Sigma_X(A-I)^\top+\Sigma_{\xi}].$$ One approach I'm trying is to replace $(A, \Sigma_{\xi})$ with $(\tilde A, I)$ with $\...
Harry556's user avatar
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2 votes
0 answers
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Prove that $\xi (\rho) = tr_{env} \left[ U (\rho \otimes \rho_{env}) U^{\dagger}\right] = \sum_{k} E_{k}\rho E_{k}^{\dagger}$

In section 8.2.3 of Nielsen and Chuang, there is a derivation of the operator-sum representation as follows: $\xi (\rho) = tr_{env} \left[ U (\rho \otimes \rho_{env}) U^{\dagger}\right] \tag{1}$ And $\...
JiQing's user avatar
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1 vote
1 answer
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trace of wedge product and cyclic property [closed]

Surprisingly, while their are similar but more advanced questions on this site, I don't see any answers to the basic version I am asking herein. If I am taking the trace of a wedge product of matrices,...
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