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Questions tagged [trace]

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

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1answer
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Summation of singular values

If $A \in \mathbb{R}^{m\times n}$, then show that $$\sum_{s=1}^{r}\sigma_s(A)=\text{max}\{\text{trace}(U^TAV): U \in \mathbb{R}^{m\times r}, V\in \mathbb{R}^{n\times r}, \text{and} \ U^TU=V^TV=I_r\}.$$...
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Finding eigenvalues of a matrix given its characteristic polynomial and the trace and determinant

I am told a matrix A has characteristic polynomial: $(\lambda−1)^3(a\lambda+\lambda^2+b),$ and that $\text {tr}(A)=12,$ and $\det(A) =14.$ I am asked to find the eigenvalues. Is the only to do this ...
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1answer
11 views

Trace of a doubly-stochastic matrix

Is there anything special about the trace of a doubly-stochastic matrix ? Formally, let $\mathbf{A}$ be doubly-stochastic of size $n$, and write $\mathrm{Tr}(\mathbf{A}) = \sum_{i = 1}^{n} \mathbf{A}...
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1answer
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What is the limit of $\mathrm{Tr}(G^kM{G^*}^k)^{1/2k}$ when $k$ goes to infinity?

If $G\in \mathscr M_n(\mathbf C)$ then it's well known that $\lim_{k\to \infty}\|G^k\|^{1/k}=\rho(G)$ where $\rho(G)$ is the spectral radius of $G$, the value of the limit does not depend on the ...
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2answers
62 views

Solving for a matrix in an equation with trace

I want to solve the following equation for the $m\times m$ matrix $X$: $$2X^T(A^TA)=-2\mathrm{trace}(X)I+V,$$ where $I$ is the identity matrix, $A$ is a $t\times m$ known matrix, and $V$ is an $m\...
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0answers
32 views

relationship between the sum of a matrix A's singular values and $max[trace(U^TAV)]$

For a $m \times n$ matrix $A$, how to show $\Sigma_{k=1}^{r}\sigma_k(A) = max\{trace(U^T A V)\}$, where $U$ is m by r, V is n by r, and $U^TU = V^TV = I$? ( $\sigma_k(A)$ are the singular values of A )...
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1answer
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$\max \Re \, \text{tr} (U V)$, with $U$ and $V$ unitary

I'm trying to find the $V$ which maximizes $$ \Re \, \text{tr}(UV) $$ where $U$ and $V$ are unitary matrices, and $U$ is given. I'm starting to expect that the solution is quite trivial, but I might ...
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1answer
30 views

Definition of trace in Bourbaki

Bourbaki, General Topology, p. 61 (1966) What is the definition of trace in the following Proposition? Proposition 8. Let $\mathcal{F}$ be a filter on a set $X$ and $A$ a subset of $X$. Then the ...
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1answer
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proof of normal equations for ordinary least squares using matrix trace

I follow the proof on Wiki. However, I first encountered this while looking at these notes from Stanford CS229. Section 2.2 contains a proof that uses matrix traces, including this part from page 11: ...
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Which relationship between trace and determinant is established using density?

I read in some lecture notes that "as an example for the intersection between linear algebra and calculus, one can establish the relationship between trace and determinant of a matrix using a density-...
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Simplifying trace of matrix product — why does this hold?

I have a question about a line in the textbook "Gaussian Processes for Machine Learning" by Rasmussen and Williams (available here). On page 125 (equation 5.23), they write the following: $-\frac{1}{...
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2answers
52 views

Unitary in the Orthocomplement of a Matrix

Given a matrix $A \in M_n(\Bbb{C})$, does there exist a unitary $U \in M_n(\Bbb{C})$ such that $tr(AU)=0$? Geometrically it seems true: thinking of $A$ as a vector in the plane, and unitaries as ...
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Showing $Tr_{L/K} \circ Tr_{M/L} = Tr_{M/K}$ where $K \subseteq L \subseteq M$ is a tower of field extensions - *Algebraic Number Theory* by Neukerich

I'm reading Algebraic Number Theory by Neukirch and I'm having trouble the proof for the statement: (2.7) Corollary: I a tower of finite field extensions $K \subseteq L \subseteq M$, one has $$...
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1answer
27 views

Unsure of how to proceed through Trace Property

I'm having difficulty figuring out how to derive the following from Andrew Ng's CS229 lecture notes. $$\nabla_A \textrm{Tr } ABA^{T}C = CAB + C^TAB^T $$ where $\textrm{Tr }$ is the trace operator ...
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Trace of the square root inside commutation property

Hi I'am looking forward to prove this statement, which seems to be true (checked numerically) even if I cannot find it anywhere and didn't manage to prove it : $\forall B\in M_{n,r}(\mathbb{R}),Tr(\...
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1answer
71 views

Trace of $x\otimes y\tilde\otimes x\otimes y$

Suppose that $x,y\in\mathbb H$, where $\mathbb H$ is a separable Hilbert space. Define the tensor $x\otimes y:\mathbb H\to\mathbb H$ by setting $(x\otimes y)(z)=\langle z,y\rangle x$ for $z\in\mathbb ...
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0answers
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weak convergence and trace operator (Neumann boundary condition)

We have $f_k \rightharpoonup f$ in $L^2(0,T;W^{1,2}(\Omega))$, where the $f_k$ are said to satisfy the boundary condition $\nabla f_k \cdot\text{n}|_{\partial \Omega} = 0$ in the sense of traces. ...
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3answers
135 views

Real matrix $A_{3\times 3}$ such that $\operatorname{tr(}A)=0$ and $A^2+A^T=I$?

I'm dealing with the test of the International Mathematics Competition for University Students, 2011, and I've had a lot of difficulties, so I hope someone could help me to discuss the questions. The ...
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1answer
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Check if the function $h(A) = \lim\limits_{n \to +\infty} \frac{\text{tr}((A^A)^{n+1})}{\text{tr}((A^A)^{n})}$ is a valid matrix norm

For a given function $$h(A) = \lim\limits_{n \to +\infty} \dfrac{\text{tr}((A^A)^{n+1})}{\text{tr}((A^A)^{n})}$$ we have to check if it's a valid matrix norm. I know that $A^A$ is defined as $$A^A ...
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1answer
71 views

Prove that $\text{tr}(((A+B)^T(A+B))^{1/2}) \leq \text{tr}((A^TA)^{1/2}) + \text{tr}((B^TB)^{1/2})$

I'm trying to show that the nuclear norm (sum of singular values of the matrix) is actually a valid matrix norm. I know that $$\sum\limits_{i=1}^n \sigma_i(A) = \text{tr}((A^TA)^{1/2})$$ So now what ...
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What matrices fulfill trace$(A^2)=0$?

I am looking for real $n \times n$ matrices such that trace$(AA)=0$. So far I have found out that $A^2=BC-CB$ for some $n \times n$ The nullspace of trace is generated by $\{E_{ij} | i \neq j\} \...
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trace norm of square root of positive definite matrix

Obviously, there is a nuclear norm inequality, i.e., $$\|[\mathbf{A}, \mathbf{B}]\|_* = \operatorname{trace}(\sqrt{\mathbf{AA'+BB'}}) \geq \operatorname{trace}(\sqrt{\mathbf{AA'}})=\|\mathbf{A}\|_*.$$ ...
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Understanding the behavior of the trace in calculating the gradient

In Andrew Ng's notes in Machine Learning, I found this equality that I don't understand why it's true. $$∇_θJ(θ) = (1/2) ∇_θ (θ^TX^T Xθ − θ^T X^T y − y^TXθ + y^T y)$$ $$ = (1/2) ∇_θ tr(θ^TX^T Xθ − θ^T ...
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Trace on $\mathcal{S}(\mathbb{R}^k) \mathbin{\hat{\otimes}_\pi} \mathcal{S}'(\mathbb{R}^k)$

$\newcommand{\Tr}{\operatorname{Tr}}$Let $\mathcal{S}(\mathbb{R}^k)$ denote the $k$-dimensional Schwartz space with the usual topology, and let $\mathcal{S}'(\mathbb{R}^{k}))$ denote its strong dual (...
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1answer
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Change the trace of a Matrix

I want to know if I have a matrix $A \in \mathbb{M}_{n\times n}(\mathbb{K}) $ and I want to change the trace multiplying it by a number $\beta \in \mathbb{K}$: $$ A=\left( \begin{array}{ccc} \alpha_{...
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Why can the covariance matrix $\mathbf W_B$ be written as $E[\mathbf s_B \mathbf s^H_B]$ and the relation between power and covariance

In this paragraph,the author said that a covariance matrix $\mathbf W_B$ can be written as the mean of $\mathbf s_B \mathbf s^H_B$,why ?i don't understand about this,and it also said that the ...
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2answers
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Proof of this trace inequality

In an article I was reading, it is written: It is well-know that for any two matrices X and Y of appropriate sizes we have $$|tr\, X^*Y| \leq \frac{tr \, X^*X + tr \, Y^*Y}{2}$$ I didn’t find it ...
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1answer
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What is the “right” extra information needed to define the trace of a map between two different vector spaces?

Let $V,W$ be real vector spaces of dimension $d$, and let $T \in \text{Hom}(V,W)$. Is there a "natural piece of additional information" required in order to give a meaningful interpretation of the ...
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1answer
59 views

Perturbation Theory: Derivative of a trace.

The problem that I am looking at is the following perturbation problem from the notes on Trace Inequalities and Quantum Entropy on page 12, the following result is said to follow "by the Spectral ...
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Trace of Sobolev functions on a circle of radius R>0

I have a difficulty dealing with a seemingly elementary question. Let $f \in H^m(K)$, $m \geq 1$, where $K$ is a (possibly large) bounded domain in $\mathbb R^2$. Let $\Gamma$ be a smooth codimension ...
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1answer
66 views

Derivative of $\mathrm{tr}((A^{1/2}BA^{1/2})^{1/2})$ w.r.t $A$

Let $A,B\succeq 0$ be two p.s.d. matrices. What is the derivative of $\mathrm{tr}((A^{1/2}BA^{1/2})^{1/2})$ w.r.t $A$?
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Geometrical interpretation of the trace? [duplicate]

A while back, I learnt about the geometrical interpretation of the determinant of a linear map, that given the exterior algebra of vector spaces $V$, and a linear map $\phi: V \to W$, then we can ...
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2answers
136 views

A matrix inequality involving trace norm of a matrix and its inverse

Let $A,B \succeq 0$ be two positive semidefinite matrices. Can we get a closed form expression for the following quantity? $$ \inf_{X \succ 0} \mathrm{tr}(XA) + \mathrm{tr}(X^{-1}B) $$ We assume all ...
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2answers
118 views

Trace of product of two Hermitian matrices

Let $A$ and $B$ be two Hermitian complex matrices. (a) Prove that $\operatorname{tr}(AB)$ is real. (b) Prove that if $A, B$ are positive, then $\operatorname{tr}(AB)>0$. (a) The trace of ...
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1answer
73 views

How to derive this matrix trace formula?

I am trying to derive equation (7) from this physics paper (sorry for the paywall) $M_n$ for $n\geq 0$ is a sequence of $2\times 2$ real matrices with determinant $1$. These matrices obey the ...
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1answer
152 views

How to prove $\rm{tr}(A)=\rm{tr}(B)$ for real $2018\times2018$-matrices such that $A^{2018}=I=B^{2018}$ and $AB=BA$ and $\rm{tr}(AB)=2018$ [closed]

Let $A,B$ are two real square matrices of order 2018 such that $A^{2018}=I=B^{2018}$ and $AB=BA$ and $\operatorname{tr}(AB)=2018$. Prove that $\operatorname{tr}(A)=\operatorname{tr}(B)$.
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When does trace and determinant of a 2 x 2 matrix equal each other? (Linear Algebra)

Background Information: I am new to linear algebra, and I recently came across this homework question that I am confused about. I appreciate any explanation that can help me improve my solution. ...
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Is there a name for the sum of the off-diagonal components of a tensor?

If you have a matrix, the "trace" of the matrix is the sum of the diagonal components of the matrix. For example, given a matrix $\mathbf{A}$: $$\mathbf{A} = \begin{bmatrix}a_{11} & a_{12} & ...
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1answer
44 views

Meaning of surface measure

While studying PDE, I came across this trace operator which talks about the class $L^{p}(\partial \Omega)$ for $\Omega \subset \mathbb{R}^{n}$ is open with $C^{1}$ boundary. I don't understand what ...
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1answer
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Weird Notation for Trace of an Endomorphism

I am having some difficulty understanding a piece of notation from Riemannian Geometry: and Introduction to Curvature by John M. Lee. In Section 2 just under equation 2.3 Lee defines the trace ...
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0answers
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Applications of the Selberg-Arthur trace formula

I would like to know about practical applications of the Selberg-Arthur trace formula. I would be specially interested in possible cases of usage in the study of formal and natural languages. Best ...
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0answers
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For a real square matrix $S$, what can be said about $Tr(S^2)$?

For a real square matrix $S$ with all eigenvalues on the imaginary axis, what can be said about $Tr(S^2)$? Since all eigenvalues of $S$ are on imaginary axis, $Tr(S)=0$ but I was wondering if anything ...
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1answer
36 views

Trace of a matrix by eigendecomposition

Let $A$ denote matrix based on another matrix $B$: \begin{align} A = (I + \lambda B)^{-1} \end{align} $I$ is the identity matrix and $\lambda$ is a coefficient. Decomposing $B$ as $USU^T$ where $U^...
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1answer
31 views

Is the exponential of the trace convex?

Take $a \in \mathbb{R}^+$ and $B,X \in \mathbb{R}^{d\times d}$ where $B=B^T$ and $X = X^T$, $a$ and $B$ are fixed. Is then the function $$f(X) = ae^{\operatorname{tr}(BX)}$$ convex in X? For $d=1$ ...
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2answers
52 views

Positive-definite, Symmetric Matrix Problem

I have a question that I've been working on for a bit now, and it says, "Let $A\in M_{2\times2}(\mathbb{R})$ be a symmetric matrix. We say that A is positive definite if all of the eigenvalues of $A$ ...
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0answers
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Is there a generalization of Araki-Lieb-Thirring inequality for four matrices?

It is known that $$ Tr[(AB)^n] \leq Tr (A^n B^n)$$ for $A$, $B$ positive semi-definite matrices. I am looking of a generalization of it for a product of $4$ positive semi-definite matrices of the the ...
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0answers
29 views

Trace ratios, ratio traces and generalized Rayleigh quotient

Reading about Linear Discriminant Analysis, I find many problem formulations, that don't seem to be equivalent. So I'm trying to find the problem formulation whose solution is expressed as the ...
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1answer
68 views

Trace of squared non-square matrix

In reading a paper I came across this expression which I don't quite understand: $$ \frac{\lambda_1}{2N}\operatorname{tr}\left((\mathbf{H}^M\mathbf{H}^M)^T\right) $$ For context, $\lambda_1$ and $...
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0answers
17 views

$\mathbf{L} \succeq \mathbf{Q}$ true? For, $\mathbf{L}$ diagonal with all entries $\leq P$ and $\mathbf{Q}$ Hermitian, with its trace limited by $P$

I have the Hermitian and positive semidefinite matrix $\mathbf{Q}$, with $\text{tr}\{\mathbf{Q}\} \leq P$ and the diagonal matrix $$ \mathbf{D} = \text{diag}(d_1, ..., d_N)\, , $$ with $$ 0 \leq d_1 \...
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1answer
53 views

Matrix trace equality [closed]

Suppose $A$ and $B$ are positive definite matrices of the same size. Prove that \begin{align} \operatorname{Tr}( A^{1/2} (A^{-1/2} B A^{-1/2})^{1/2} A^{1/2}) \leq \operatorname{Tr}( (A^{1/2} B A^{1/2}...