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.
1,626
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Using $(e^{ik\cdot x})$ as an orthonormal basis for $L^2(\mathbb{R}^d;\mathbb{C})$ to define trace.
I know that $e^{ik\cdot x}$ are not elements of $L^2$. But I believe this is often used in quantum mechanics, and wondered if there is some justification for it.
For example, I have seen the trace of ...
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Trace of off-diagonal blocks of a positive semidefinite matrix
Consider the matrix
$$A=\begin{pmatrix} A_1 & A_2 \\ A_3 & A_4 \end{pmatrix}$$
Let's suppose that $A$ is a real $n\times n$ positive semidefinite and satisfies $\|A\|\leq 1$, i.e., the largest ...
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Understanding proof of convexity of trace function
I am trying to understand Theorem 2 ($F(A,K) := \text{tr}(A^{-r}K^{\ast} A^{-p} K)$ is convex for $p,r \geq 0$ and $p+r \leq 1$) of the paper Convex trace functions and the Wigner-Yanase-Dyson ...
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Show that there exists a $λ ∈ K$ such that $l(A) = λtrace(A)$ for all $A ∈ Mat_{n,n}(K)$
Hey I want to check my solutions with this problem. Can someone help me?
Let $K$ be a field, $n ∈ \mathbb{N}$, and $l: Mat_{n,n}(K) → K$ a linear form such that $l(AB) = l(BA)$ for all $A, B ∈ Mat_{n,...
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What is the trace of the second tensor component?
My teacher asks to prove that for any matrix $A \in \operatorname{Mat}(N, \mathbb{C})$ there is true:
$$
A=\operatorname{tr}_2\left(P_{12} A_2\right)
$$
where $A_{2}=E_{N}\otimes A$ and $P_{12}$ is ...
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Is the trace of a constant simply that constant?
I can't seem to find an answer to this online, basically what I am asking is
$$\mathrm{Tr} ((4)_{1\times 1})=4?$$ or in general is $\mathrm{Tr} ((k)_{1\times 1})=k?$ Where $k$ is a constant.
I think ...
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Bound on the trace norm of a matrix written in a continuous basis
For a generic trace-class matrix $A=\sum_{n,m}A_{n,m}|n\rangle\langle m|$, one can easily find the bound $\|A\|_1\leq \|A\|_{1,1}$, where $\|A\|_1=\sum_{k}\sigma_k(A)$ is the trace norm and $\|A\|_{1,...
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Differential of the Trace for functions applied to unbounded self-adjoint operators
Let $f\colon\mathbb{R}\to\mathbb{R}$ be in the set of Schwartz functions (or any functions with nice enough integrating properties on the real axis), $A,M$ two (possibly unbounded) self-adjoint ...
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prove that ${\rm tr}(\sqrt{\sqrt{AA^T}BB^T\sqrt{AA^T}})={\rm tr}(\sqrt{B^TAA^TB})$
Suppose $A,B\in \mathbb{R}^{m\times n}$, prove that ${\rm tr}(\sqrt{\sqrt{AA^T}BB^T\sqrt{AA^T}})={\rm tr}(\sqrt{B^TAA^TB})$.
The form of LHS comes from fidelity in quantum information (see this Prove $...
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Gradient of $X \mapsto \mbox{tr} \left(BXCX^TB^TBXCX^TB^T\right)$
Let us assume that
\begin{equation}
f(X)=\mbox{tr}\left(XCX^TXCX^T\right),
\end{equation}
in which $C\in\mathbb{R}^{r\times r}$ is a symmetric matrix, and $X \in \mathbb{R}^{r'\times r}$. From the ...
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Solving a matrix equation involving trace and determinant
Let $n\in\mathbb{Z}_{>1}$. Consider the following set :
$$
E_n=\{A\in\mathcal{M}_n(\mathbb{C})| \mathrm{Tr}(A)=\det(A)\}
$$
My goal is to characterize $E_n$ for any $n$.
I have already found a ...
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Minimize $\operatorname{tr} \left(X^T L X \right)$ subject to inequality constraints
Given a matrix $Y$, a symmetric matrix $L$ and a binary matrix $B$, I would like to solve an optimization problem of the form
$$ \min_{X} \quad \operatorname{trace} \left( X^T L X \right) \quad\quad \...
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Derivative of the function of a trace with respect to a vector
I have the following function of a trace
$$
f(\mathbf{x})=\mathrm{tr}(AM(\mathbf{x})A^T)
$$
with $A$ and $M$ two $n\times n$ square matrices, and $\mathbf{x}$ a vector in $\mathbb{R}^m$. The matrix $M$...
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Simplify ${\rm trace}((A_1⊗A_2+B_1⊗B_2)^{-1})$
This equation is take a long time in simulation due to Kronecker product ($⊗$).
Simplify $${\rm tr}((A_1⊗A_2+B_1⊗B_2)^{-1})$$
where tr is trace operator (sum of diagonal elements).
Also, $A_1,A_2,B_1,...
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Smallest constant $C$ to bound $\mathbb{E}[\mathrm{tr}((\overline{X}_n + I)^{-1})] \leq C~\mathrm{tr}((\mathbb{E}[X] + I)^{-1})$?
Let $X_1, \dots, X_n$ be random real-valued symmetric rank-one matrices,
$$
X_i = x_i \otimes x_i,
$$
where $x_i$ are such that the standard Euclidean norm satisfies $\|x_i\|^2 \leq a$ almost surely.
...
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Asymptotics of traces of powers of Gaussian kernel on bounded domains (ball)
Let $K({\bf r}_1,{\bf r}_2)= e^{-\frac12|{\bf r}_1-{\bf r}_2|^2}$ be the translation and rotation invariant Gaussian kernel. I want to compute the trace Tr$\hspace{1pt}K^3$ over the $n$-dimensional ...
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Largest $k$ such that $k \operatorname{Tr} H^2 C \le (\operatorname{Tr} H C)^2$
Suppose we are given $H$, a positive semidefinite $d\times d$matrix. How do I find $k$, the largest $k'$ such that the following holds for all positive semidefinite $C$ with unit trace?
$$k' \...
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Trace inequality of Hadmard product
For any Hermitian matrix $\bf A$ and invertible matrix $\bf B$, how do I derive the inequality below
$${\rm tr}(({\bf B}^H{\bf A}{\bf B}) \circ({\bf B}^H{\bf A}{\bf B}))\geq \lambda_{\min}^4({\bf B}){\...
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bounds on $\frac{(\operatorname{Tr} H C)^2}{\operatorname{Tr} H^2 C}$
Suppose $H,C$ are positive semi-definite matrices, and $C$ has unit trace. $H$ is given and fixed while $C$ is allowed to vary. How do I bound the following quantity?
$$R=\frac{(\operatorname{Tr} H C)^...
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Trace of PSD matrices [closed]
Let matrices $A, B \in \mathbb{R}^{n \times n}$ be positive semidefinite (PSD). Let $B = V V^T$, where $V \in \mathbb{R}^{n \times n}$. Why does the last equality hold in the following?
$$ \mbox{tr} (...
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Property of trace of a matrix
I am trying to understand a solution to a problem and stuck at the step where it mentions
$$ \sum^n_{k=1} x^T_ky_iy^T_ix_k = tr(x^Ty_iy^T_ix)$$ , $$ x_k,y_i \in \mathbb{R^d}$$
I understand that the ...
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Upper bound of a matrix using trace
I was reading a proof that uses the following matrix relation;
$$
A \leq \text{Tr}(A)I
$$
Where $\text{Tr}(\cdot)$ denotes the trace operator and $I$ is the identity matrix of appropriate dimension.
...
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Is it possible to get the trace value of this matrix?
I want to find a general formula for the trace of the following $N\times N$ matrix raised to the power of $d$, where $d \in \mathbb{N}$.
$$
\begin{bmatrix}
N-2 & 1 & 0 & \cdots & 0 &...
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For positive semidefinite unit trace $\Gamma$, what is $\{ x \in \mathbb R^N: x^T A x = \text{tr}(A \Gamma) \ \forall A \in\mathbb R^{N \times N} \}$?
My question.
Let $N > 1$ and $\Gamma \in \mathbb R^{N \times N}$ be a positive semidefinite matrix with unit trace.
Can we describe in a simple manner the following set? $$S(\Gamma) := \{ x \in \...
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Trace of product of orthogonal projectors is an inner product $\operatorname{tr}(\pi_{x'}\pi_{y'}) = |\langle y,x\rangle|^2$
I am reading something in quantum mechanics, and it has the following claim.
Setting: Consider a complex vector space $\mathbb{C}^k$, its projective space $P(\mathbb{C}^k)$. For nonzero $x \in \mathbb{...
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Spectral triple for a (real) full matrix algebra
Let $\mathcal A = \mathbb R^{N \times N}$ be the real full matrix algebra, $N \in \mathbb N_{> 1}$, which is represented by the Hilbert space $H := \mathbb R^N$ (that is, $\mathcal A \to B(H)$, $A \...
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Lim trace of inverse positive definite matrices
We have a symmetric positive definite matrix $A \in \Bbb R^{m \times m}$. How to calculate the following?
$$ \lim_{n \to \infty} \mbox{tr} \big( (\underbrace{A\cdot A\cdots A}_n)^{-1} - (\underbrace{A\...
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Why does the sum of eigenvalues equal to trace in terms of linear transformations?
While studying eigenvectors, I was confronted with two statements:
The product of the eigenvalues of some matrix $A$ is equal to the determinant of $A$
The trace of $A$ is equal to the sum of its ...
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Does $\operatorname{Tr}(AB) \ge 0$ imply $A \ge 0$?
Consider a fixed matrix $A \in \Bbb C^{n \times n}$ and the $\Bbb C^{n \times n} \to \Bbb C$ function $B \mapsto \operatorname{Tr}(AB)$. If $A$ is positive definite, then I know that $\operatorname{Tr}...
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PCA proof, trace move
In the PCA proof, I see this move:
What is the "trace" algebra/rule that allow this trace-move?
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Commutative property for Hilbert-Schmidt norm?
Let $A,B$ be linear, compact, self-adjoint and even trace-class operators.
Can I bound $\|ABA^{-1}\|_{HS}$ by the norm of $\|B\|_{HS}$ somehow? (Where the bound does not depend on $\|A^{-1}\|$.)
Here ...
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$\sigma$-Weakly Continuous Bounded Linear Functional on a von Neumann Algebra is Normal
I have been working on Exercise 4 in Chapter 4 of Murphy's "$C^*$-Algebras and Operator Theory", which is as follows:
Let $A$ be a von Neumann algebra on $H$, and suppose that $\tau$ is a ...
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If $X$ is a diagonal matrix of $(N \times N)$ size, then what will be the $ tr ( \frac{1}{X+\frac{1}{z}I})$?
I am trying to invert $\eta$-transform (or N-transform) numerically, for which I have $\eta(z) = \frac{1}{Nz} tr( \frac{1}{X + \frac{1}{z}I})$. But I am not sure how I can find $\eta^{-1}(z)$, as I am ...
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How to solve the problem of trace optimization which includes Hadamard product?
I have the following minimization problem, where I want to find W,
\begin{align}
&\min \mathrm{tr} (((W^TK)\circ(W^TK))^T((W^TK)\circ(W^TK))L)\\
&\text{s.t.} ~ W^TKHKW = I
\end{align}
where $\...
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Prove the following equation about the expension of the determinant of an operator
About the expansion of the determinant of an integral operator $K:=\phi(x)\rightarrow \int_a^b K(x,y)\phi(y){\rm d}y$
${\rm det}(1+K) = \sum_n \frac{\alpha_n}{n!}$, where $\alpha_n = \int \cdots \int {...
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What does this notation (inner product with differential) mean?
The following symbols are used in On optimal transport of matrix-valued measures by Yann Brenier and Dmitry Vorotnikov (cf. p. 3):
$A : B := \text{tr}(A B^{\mathsf{T}}) = \sum_{i, j = 1}^{d} a_{i, j} ...
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Explicit form for the total variation measure of a symmetric matrix-valued measure
Let $X$ be a locally compact Polish space.
Consider the set $M(X; S^d)$ of real symmetric matrix-valued measures on $X$, that is, the set of countably (with respect to the Frobenius norm $\| \cdot \|...
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Finding the missing values in a matrix given eigenpairs
Find the missing values of this 3x3 matrix A,
$$A=\begin{pmatrix}-3 & a & b\\\ c & 1 & d \\\ e & f & -1\end{pmatrix}$$
given the eigenpairs
$$λ_1 = -2, x_1 = \begin{pmatrix}-1 \...
user1136916
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How to find the eingenvalues of matrix $M$ with $\text{rank}(M)\leqslant1\;?$
If a $\,n\times n\,$ matrix $M$ satisfies
$\text{Rank}(M)\leqslant1\;,\;$ then $\;\det(I+tM)=1+t\text{Tr}(M)\;,$
where $\;\text{Tr}(M)\;$ denotes the trace of $\,M$.
In this book it is stated that all ...
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1
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Implicit assumed existence of trace?
Morning!
In pde often there are boundary conditions for Sobolev functions. For example, "If $f\leq g$ on $\partial E$, then..." where $E$ is a domain in $\mathbb{R}^n$ and $f,g\in W^{1,p}(E)$...
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Stochastic heat equation
If we consider the stochastic heat equation and the Laplacian as a generator of the $C_0$-semigroup $S(\cdot)$ given by the kernel heat, how can I show that we have
$$Tr[S(r)S^*(r)] <\infty.$$
I ...
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Exercise 9, Section 6.3 of Hoffman’s Linear Algebra
Let $A$ be an $n \times n$ matrix with characteristic polynomial $$f=(x-c_1)^{d_1}\cdots (x - c_k)^{d_k}$$ Show that $$c_ld_1+\cdots + c_kd_k=\text{trace} (A)$$
My attempt: Characteristic polynomial ...
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Trace -logarithm - matrix
How do I calculate this, $\beta$ is a parameter, $H$ a matrix:
$(1-\beta \partial_\beta)\ln(Tr(e^{-\beta H})) = \ln(Tr(e^{-\beta H})) - \beta \frac{\partial_\beta Tr(e^{-\beta H})}{Tr(e^{-\beta H})}=?$...
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find an orthogonal basis and signature of this two trace based inner prodcuts
find an orthogonal basis and signature for each of following inner products:
$g(A,B)=tr(A^TUB)$
$h(A,B)=tr(AUB)$
$U=\begin{bmatrix}2 & 1 \\
1 & 1
\end{bmatrix}$
firstly tried with g and i ...
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1
answer
38
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Why is the trace of matrices in any power a Casimir function for the Poisson structure on $\mathfrak{gl}(N)$?
Let's assume we have a Poisson bracket
$$
\{S_{ab}, S_{cd}\} = S_{cb} \delta_{ad} - S_{ad} \delta_{cb} \,.
$$
It's obvious that $C_k = \mathrm{tr}(S)$ (where $S = \sum_{ij} S_{ij} E_{ij}$) is a ...
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Stochastic Integrals, Inverses, and Trace
I have an even, positive semi-definite function in $L^2([0,1] \times [0,1])$, where by the Spectral theorem
$$k(r,s) = \sum_{n = 1}^\infty \lambda_n f_n(r) f_n(s)$$
where $\lambda_n$ is a eigen value, ...
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Basis-free definition of trace class operators?
Let $H$ be a separable complex Hilbert space. If $\dim(H)<\infty$, then the trace of every operator $T$ is well defined, and it can be computed from the matrix of $T$ in any basis. Although this ...
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Reference for the upper bound of trace of product of matrices
For two real-valued square matrices $A$ and $B$, it has been shown that (for example, in this post),
$$ tr(AB) \leq \lVert A \rVert Tr(\sqrt{B^{\prime}B}).$$
I am looking for a reference about this ...
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Simultaneous traceless matrices decomposition
Let us say that we have two $4 \times 4$ complex, traceless matrices $A, B$. As $A, B$ are traceless we can rewrite them as
$$
A = XY - YX \\
B = UV - VU
$$
How strong relationship between those ...
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2
answers
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Trace of an indefinite operator does not exist
I see that there are a few definitions of trace-class operator. They are roughly the same, but some include a clause that the operator should be definite in some flavor. I want to ask why this clause ...
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