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Questions tagged [trace]

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

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29 views

Prove that $ \mbox{Tr}(AB)\leq \sum_{i=1}^n \lambda_{i}(A) \lambda_i(B)$, where $A, B$ are $n \times n$ Hermitian matrices

Suppose $A, B$ are $n \times n$ Hermitian matrices, i.e., $ A^{T}=\bar{A}$ and $B^{T}=\bar{B} $, prove that $$ \operatorname{Tr}(AB)\leq \sum_{i=1}^n \lambda_{i}(A)\lambda_i(B), $$ where $ \...
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1answer
50 views

Prove that $r(A)=\operatorname{tr}(A^2)$

Let $A\in M_n(\mathbb{C})$. Show that if $A^3=A$, then $r(A)=\operatorname{tr}(A^2)$. Since $A^3=A$, the possible eigenvalues are $0,1,-1$. I don't know from here how to compute the rank of $A$. ...
2
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1answer
13 views

Equality of tracial states on dense $C^*$-subalgebra implies equality on generated von Neumann algebra?

Maybe this is a simple question, but I'm not sure about the following: Let $\cal M$, $\cal N$ be von Neumann algebras and $X\subseteq \cal M$ a weakly dense (possibly separable) $C^*$-subalgebra. Let ...
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1answer
25 views

Derivative of a Function of the Diag function

Suppose there is a vector $U \in \mathbb{R}^n$. How would you find the derivative of: $$ F(U)=trace\left(diag(U) A\ diag(U) \right) $$ where $A \in \mathbb{R}^{n \times n} \succ 0 $ and where $diag(\...
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1answer
21 views

tracial state of a orthogonal projection

Suppose $A\in M_n(\mathbb{C})$,$A$ has eigenvalues$\lambda_1,\cdots,\lambda_n$,$P$ is the orthogonal projection from $\mathbb{C}^n$ onto the span of eigenvectors associated with $\lambda_1,\cdots,\...
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3answers
57 views

$\det(A^2+A-I_2)+\det(A^2+I_2) = 5$

Let $A \in M_{2\times 2}(\mathbb{C})$ and $\det(A)=1\DeclareMathOperator{\tr}{tr}$ Prove that $\det(A^2+A-I_2)+\det(A^2+I_2) = 5$ using Cayley-Hamilton Theorem $A^2-\tr(A)A+\det(A)I_2=0$ $\det\big(\...
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0answers
36 views

Show $ f'(0)=\det(A)\operatorname{tr}(A^{-1}B)$ [closed]

Let $A,B \in \mathbb{M}_{n\times n}(\mathbb{R})$ be square matrices with real coefficients, and consider the function $$ f(t)=\det(A+tB) $$ Show that $f$ is a polynomial in $t$, and that for ...
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4answers
232 views

Trace of symmetric matrix equals sum eigenvalues

I need to show that if $\mathbf{S}$ is symmetric, then it's trace sums to the sum of the eigenvalues. But I don't know how to show this. Can anybody give me a hint? P.S. Shame on my google skills, ...
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1answer
32 views

Proof of log identity for positive definite matrices

On Wikipedia, it is claimed that $\rm{tr}(\log(AB)) = \rm{tr}\log(A) + \rm{tr}\log(B)$. The result is valid only if $A$ and $B$ are positive definite. I use this result in entropy calculations but ...
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47 views

How can maximizing $x^T A x$ where $A$ is positive semi-definite be reduced to maximizing $trace(x^T A x)$?

Suppose $A$ is a given matrix of shape $n \times n$, and $x$ is some unknown matrix of shape $n \times m$, the objective is $$ \begin{array}{rl} & \max_x x^T A x \\ \text{subject to } & \...
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2answers
1k views

Proof for Cauchy-Schwarz inequality for Trace [closed]

Cauchy-Schwarz inequality applied to Trace of two products $\mathbf{Tr}(A'B)$ has the form $$ \mathbf{Tr}(A'B) \leq \sqrt{\mathbf{Tr}(A'A)} \sqrt{\mathbf{Tr}(B'B)} $$ I saw many places where people ...
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1answer
58 views

What is the orthogonal complement of $H^1_0$ in $H^1$?

Let $\Omega$ be a closed domain with smooth boundary in $\mathbb{R}^n$. Let $H^1_0(\Omega)$ be the closure of compactly supported smooth functions under the norm $\|u\|_1 = \int_\Omega u^2 + |\nabla u|...
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1answer
61 views

Relation between trace of matrix $A^*A$ and invertibility of the matrix $A$

Is there any relation between $\mathrm{tr}(A^*A)$ and invertibility of the matrix $A$? What information about the matrix $A$ does $A^*A$ gives ? I was confused about this when I came across the ...
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0answers
27 views

Minimize trace of $A$ given that $A−N$ is positive semi-definite and $A$ is diagonal

\begin{array}{ll} \text{minimize} & \mbox{tr} (\mathrm A)\\ \text{subject to} & \mathrm A - \mathrm N \succeq \mathrm O_n\end{array} where $A$ and $N$ are pd matrices, and $A$ is diagonal. ...
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0answers
26 views

Maximizing the trace of product of matrices under fixed spectrum

Is it correct that under fixed spectrum, $\operatorname{tr}(AB)$ is maximized when $A$ and $B$ share the same eigenbasis? If yes, how can this be shown?
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1answer
31 views

Question about trace of matrix

In literature, I found the following identity. Unfortunately, I fail to see why this holds. Where does the trace of the matrix come from? Any clarifications are highly appreciated!
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3answers
47 views

Given symmetric semidefinite matrix A and B, prove AB = 0 if and only if tr(AB)=0 [closed]

Given $A\in S^{n}_{+}$ means that $A$ and $B$ are symmetric semidefinite matrix. Can we prove that $\operatorname{tr}(AB)=0$ if and only if $AB=0$ ?
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1answer
34 views

Question on Sobolev extension onto boundary

Let $U \subset \mathbb R^3$ be an open, bounded and connected set with a $C^2-$regular boundary $\partial U$. I'm trying to understand the following implication: If $f\in W^{2-1/2,2}(U)$ then $f{\...
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1answer
14 views

For a positive semi-definite $d\times d$ matrix $A$, $ (\text{det}(AS))^{\frac{1}{d}}\leq\frac{1}{d}\text{Tr}(AS) $ for every $S\in\text{SPD}_{d}$.

For a positive semi-definite $d\times d$ matrix $A$, $$ (\text{det}(AS))^{\frac{1}{d}}\leq\frac{1}{d}\text{Tr}(AS) $$ for every $S\in\text{SPD}_{d}$. I would like to show the above statement. If ...
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1answer
28 views

Trace inequality with matrix square-roots

Suppose I have symmetric matrices $A \in \mathbb{R}^{n \times n}$ and $B \in \mathbb{R}^{n \times n}$ which are both positive definite. I am wondering if I one can bound ${\rm tr}\left(A - B \right)$ ...
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0answers
24 views

mathematical notation for the sum of all the elements of the individual diagonals above the secondary diagonal.

What is the notation of the sum of the elements above the secondary diagonal for a generic nxn square matrix? I do not mean the sum of all the elements above the secondary diagonal, but rather the ...
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3answers
61 views

Does $P^{-1} A P = P P^{-1} A$?

I hung up on the following step in a derivation I'm following and could use some guidance as to why it's true. I'm trying to show that the trace of a matrix is invariant under any similarity ...
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1answer
73 views

Number of Solutions to the Problem: Minimize $|| A - Afc^T / c^T f ||^2$ such that $\sum_i c_i = 1$

Let $A$ be a $M$ by $N$ matrix, $f$ is a column matrix with $N$ elements, $c$ is a column vector with $N$ elements that I need to solve for, and $|| \cdot ||$ is the Frobenius norm of the matrix. I ...
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1answer
69 views

How to prove that a function belongs (or does not belong) to $H^{\frac12}_{00}$

Given a domain $\Omega$, the space $H^{1/2}(\partial \Omega)$ can be defined as the image of the trace operator $\gamma: H^{1}(\Omega) \rightarrow H^{1/2}(\partial \Omega)$, which (roughly speaking) ...
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3answers
53 views

Show that there exists a matrix with trace in the set $\{1,2,…,n\}.$

Let $n$ be a positive integer and let $U$ be a finite subset of $M_{n\times n}(\mathbb{C})$ which is closed under multiplication of matrices. Show that there exists a matrix $A$ in $U$ satisfying $\...
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1answer
71 views

Derivative of a trace of quadratic form

I think the question is close to Derivative of trace of inverse of a matrix function I have a function: $f(X) = trace(X^TAX)$ And I am trying to derive $df/dX$ Update: I know that, for function $g(...
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1answer
83 views

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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2answers
72 views

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
21 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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2answers
224 views

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
77 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
37 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
20 views

$\max \Re \, \text{tr} (U V)$, 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
36 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
36 views

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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1answer
49 views

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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28 views

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
57 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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1answer
126 views

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
29 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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0answers
35 views

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
73 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
31 views

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
143 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
47 views

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 ...
2
votes
1answer
79 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 ...
3
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0answers
66 views

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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0answers
41 views

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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votes
1answer
34 views

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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0answers
93 views

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 (...