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

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16 views

Trace map from $H^1$ into $H^{\frac 12}$, does this statement imply another?

Consider trace map $T:H^1(\Omega) \to H^{\frac 12}(\partial\Omega)$ on a sufficiently smooth domain $\Omega$. It has a partial inverse $E$. If we have the statement $$F(u,Eu) = 0\quad\text{for all ...
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1answer
38 views

Proof of this theorem: $ tr(A^-)=\sum_{i=1}^r \lambda_i^{-1} $

If $A$ is an $n\times n$ symmetric matrix with $r$ nonzero characteristic roots $ \lambda_1,\lambda_2,...,\lambda_r$ then $$ {\rm tr}(A^-)=\sum_{i=1}^r \lambda_i^{-1}. $$ Note: $A^-$ is generalized ...
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0answers
45 views

$A$ be a $10\times 10$ matrix over $\mathbb R$ such that sum of each row is $1$. [on hold]

Let $A$ be an invertible $10\times10$ matrix over $\mathbb R$ such that sum of each row is $1.$ Then which option is correct? A. The sum of the entries of each row of the inverse of $A$ is ...
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3answers
42 views

Find the dimension of the space of $4\times 4$ real matrices with zero trace

I'm wondering if someone can help me to understand this problem. If $S$ is the subspace of $M_{4,4}(\mathbb{R})$ consisting of all matrices with trace $0$, what is $\dim(S)$? I've created a matrix ...
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2answers
49 views

Derivative of a trace w.r.t matrix within log of matrix sums

I'm trying to solve an optimization (sub)problem and am running into trouble with a tricky derivative. I'd like to find the matrix $C \in \mathbb{R}^{n\times d}_+$ which minimizes the following ...
2
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1answer
60 views

Prove $tr(\mathbf{A} + \mathbf{B} ) = tr(\mathbf{A}) + tr(\mathbf{B})$

Supposedly, this is an easy proof. But I'm really inexperienced and have little mathematical sophistication (trying to improve). Prove $tr(\mathbf{A} + \mathbf{B} ) = tr(\mathbf{A}) + tr(\mathbf{B})$ ...
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0answers
12 views

Triangle inequality for state distinguishability

I'm trying to understand the proof for the triangle inequality for state distinguishably in quantum theory. It is given as $\delta(\rho,\sigma) \leq \delta(\rho,\tau) + \delta(\tau,\sigma)$ with ...
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1answer
13 views

Using Eigenvalues to prove a matrix?

In regard to eigenvalues and eigenvectors in Linear Algebra, How can I prove that the characteristic equation of a $2 \times 2$ matrix $A$ can be expressed as $$ \lambda^2- tr(A)\lambda + \det(A)=0 ...
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0answers
26 views

Traces of $W^{1,\infty}$ functions

Let $\Omega$ be a Lipschitz domain and $p \in (1,\infty)$. It's known that if $u \in W^{1,p}(\Omega)$ then $u_{|\partial \Omega} \in W^{1-\frac{1}{p},p}(\partial \Omega)$. I'm wondering if the ...
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0answers
19 views

What's wrong with the following trace optimization?

I'm reading a paper that has used the augmented Lagrange function for optimization. I've tried to derive one subproblem but got a different answer from that in the paper. Could you help check it ...
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1answer
18 views

prove characterstic polynomial of $2\times 2$ matrix is $C_{A}(x)=x^2-(\lambda_{1}+ \lambda_{2})x+\lambda_{1} \lambda_{2}$

Let $$ A=\begin{bmatrix} a_{11} & a_{12}\\ a_{21} & a_{22}\\ \end{bmatrix}$$ Let $\lambda_{1}, \lambda_{2}$ not necessarily distinct, be the eigenvalues of A. Show that $$ ...
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1answer
17 views

Is it right for chain rule in trace function?

The objective function is $$ f(X)=\min_X trace(B^TX^TCXBD) $$ we know the following derivatives from Matrix Cookbook, $$ \frac{\delta{trace(B^TX^TCXB)}}{\delta X}=C^TXBB^T+CXBB^T \\ \frac{\delta ...
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0answers
12 views

Representing the sum of squared diagonal elements of a matrix with trace function

Assume that we have a $n\times n$ vector called $A$. I am interested in computing $\sum_{i=1}^n A_{ii}^2$ (i.e., sum of the squared diagonal elements). However, I want to do so as trace function and ...
3
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2answers
44 views

Proof that the characteristic polynomial of a $2 \times 2$ matrix is $x^2 - \text{tr}(A) x + \det (A)$

Let $$ A=\begin{bmatrix} a_{11} & a_{12}\\ a_{21} & a_{22}\\ \end{bmatrix}$$ Let $C_{A}(x) := \det(xI-A)$ be the characteristic polynomial of A. Show that ...
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1answer
38 views

Trace of a matrix

"the preceding 2 scalar solutions correspond to the vector solutions $x^1(t)=(t,1)^T$ and $x^2(t)=(t^2,2t)^T$ which have the Wronskian $$W(t)=\det\left[\begin{array}{lr} \mbox t & t^2\\ \mbox 1 ...
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1answer
22 views

Symplectic structures on Hermitian matrices

This is a question taken from Ana Cannas da silva's book on symplectic geometry. Let $\xi\in\mathcal{H}$, the vector space of $n\times n$ hermitian matrix. Define ...
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0answers
9 views

Trace of Log of Anti-diagonal matrix

Is it true in general that the trace of log of an anti-diagonal matrix equals to the sum of log of anti diagonal elements? For a definite example, I have encountered a problem in which I need to ...
0
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0answers
29 views

What's the relations between trace and Frobenius norm of a matrix?

Let $X=U^T\Lambda V$, then we have $tr(X)=\sum_i|\lambda_i|$ and $||X||_F=\sqrt{\sum_i\lambda_i^2}$, then we could get $||X||_F \leq tr(X)$. Is the following inequality correct? $$ tr(A^TBC) \leq ...
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1answer
32 views

Trace and transformations of a matrix

I have the following expression: T = Trace{AMA'} Where M is a square nxn matrix, and A is a nxm matrix, both both full-rank. The goal I want to acheive is that I do not want any of the off-diagonal ...
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1answer
30 views

Trace norm identity (in bra-ket notation)

I came across the following identity in a paper: $$ \|\hspace{0.3em}|v\rangle\langle v| - |w\rangle\langle w|\hspace{0.3em}\|_{tr}=2\sqrt{1-|\langle v|w\rangle |^2}$$ where the norm on the left is ...
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1answer
25 views

Lie group as a subset of its Lie algebra

Consider a (possibly infinite-dimensional) Lie group $\mathcal{G}$ and let $\mathcal{A}$ be an algebra with a product $\cdot$ and the bracket $[u,v]=u\cdot v - v\cdot u$. The following statement is ...
2
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1answer
70 views

A hard exercise on endomorphisms and determinants

The following exercise has been bugging me for some days, could someone help me with it ? Let $E$ be a $\mathbb{C}$-vector space with dimension $n$ and $f\in\mathcal{L}(E)$ ($\mathcal{L}(E)$ denotes ...
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1answer
34 views

$\operatorname{tr}(AB) = 0$ for (skew-)symmetric matricies

I know if A is symmetric and B is skew-symmetric then $\operatorname{tr}(AB) = 0$. (This follows because $\operatorname{tr}(AB) = -\operatorname{tr}(AB) $) Is the converse of that true? In other ...
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0answers
26 views

Equivalent definitions of the trace of a Hilbert-Schmidt operator

I am currently reading the book Spectral Methods in Automorphic Forms, and Iwaniec defines the trace operator in a different way than I am accustomed to. Throughout, assume that everything converges ...
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2answers
35 views

Proving a theorem about trace of matrix which involving generalized inverse matrix

can you prove that theorem for me: Let A be mxn matrix of rank r then, $\ tr[I-A(A'A)^-A'] = m-r $  .   $\ A' $(transpose of A) ,$\ A^- $(generalized inverse of A)
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2answers
48 views

Proof that theorems about trace of matrix :

Can somebody help me about proofs of this theorems A is an nxn matrix and $\ A^2$ = mA then, tr(A) = m rank(A) . A is an nxn matrix and k is a positive integer then, tr($\ A^k$) = $\sum_{i=1}^n ...
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3answers
47 views

Trace of a power of a matrix product

Suppose I have two 2x2 matrices $A$ and $B$. What can I say about $Tr(A^k B^k)$ versus $Tr((AB)^k)$? I know that if there is some cyclic permutation that takes $A\cdot A\cdots A B\cdot B\cdots B$ to ...
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1answer
43 views

Partial derivative of the trace of matrix entry-wise exponential?

Just checking my math here and getting some help for the exponential part. $\renewcommand{\v}[1]{\mathrm{vec}\left(#1\right)} \renewcommand{\m}[1]{\mathbf{#1}} ...
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1answer
43 views

Finding trace of a matrix $A $ such that $A^3=A$ [closed]

If $A$ is a real $n \times n$ matrix satisfying $A^3 = A,$ then Trace of $A$ is always 1) $n$ 2) 0 3) $−n$ 4) an integer in the set $\{−n,−(n − 1), \dots ,−1, 0, 1, \dots , n\}.$ how to solve ...
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1answer
64 views

Finding the trace of $(I + \Sigma^{-1} AA^T)^{-1}$

I need to efficiently compute the trace of $$ B = (I + \Sigma^{-1} AA^T)^{-1} $$ where $\Sigma$ is diagonal and all its elements strictly greater than zero. $A$ is $-1$ on the diagonal and $1$ right ...
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2answers
65 views

Is this Determinant and Trace identity equivalent to Unitary matrix?

Thanks for any help in advance. I have this equality for a 2x2 invertible complex matrix: $$\text{Tr}(AA^*)=2|\text{det}(A)|^2$$ where $*$ is complex conjugate transposition. Is this equality ...
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3answers
127 views

Positive semi-definite matrix has a non-negative trace?

A simple question: If $A$ is a positive semi-definite matrix ($A\succeq 0)$, does it imply $Tr(A)\geq 0$, where the $Tr(\cdot)$ denotes the trace. If not, any counter-example? Thanks.
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1answer
41 views

Equation involving a partial trace

Is there, in general, a solution to the following equation? $\text{Tr}_{V_1}(A(X\otimes I_{V_2})) = B$ where A is an operator on $V_1\otimes V_2$, $B$ is an operator on $V_2$, $I_{V_2}$ is the ...
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0answers
18 views

Inclusion of commutators on classical pseudodifferential operators

We denote by $Cl^\mu$ the class of classical pseudo-differential operators of order $\mu$. Consider the notation $$[Cl^{a},Cl^{b}]\hookrightarrow [Cl^{a'},Cl^{b'}]$$ which means that a commutator on ...
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1answer
52 views

Proof of Hopf's Lemma

This is a segment of the proof to "Hopf's Lemma," from page 348 of PDE Evans, 2nd edition. I have a question regarding this, at the bottom of this post. Proof. 1. Assume $c \ge 0$. We may as well ...
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1answer
63 views

Show that the trace class operators on a Hilbert space form an ideal

Let $(H, (\cdot, \cdot))$ be a separable Hilbert space over $\mathbb{L} = \mathbb{R}$ or $\mathbb{C}$. Suppose that $\{\phi_n\}_{n=1}^\infty$ is an orthonormal basis for $H$. Let $\mathcal{B}(H)$ ...
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1answer
149 views

Derivative of determinant of symmetric matrix wrt a scalar

For a given square symmetric invertible matrix $\mathbf{X}$ and scalar $\alpha$ (such that the entries of $\mathbf{X}$ depend on $\alpha$), I would like to use the following well-known expression for ...
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0answers
19 views

Traces and linear combinations of idempotents

Let $R$ be a commutative ring and $S$ a subring, and consider the following conditions on an $n$ by $n$ matrix $A$ with entries in $R$: (a) $A$ is an $S$-linear combination of idempotents of ...
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0answers
11 views

Reference request: result concerning Leray trace

Let $V$ be a vector space (possibly of infinite dimension). For a linear homomorphism $f\colon V\to V$ define $$N(f)=\bigcup_{n\in\mathbb{N}} \operatorname{ker}(\underbrace{f\circ\ldots\circ ...
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1answer
48 views

Is the trace of an idempotent matrix a sum of idempotents?

Let $R$ be a commutative ring, $n$ a positive integer, and $A$ an idempotent $n$ by $n$ matrix with entries in $R$. Is the trace of $A$ necessarily a sum of idempotents of $R$?
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21 views

Extending a trace on algebra to a trace of systems of algebras

Suppose, we have a trace $\tau$ on some algebra $\mathcal{A}$, i.e. $$\tau(aA+bB)=a\tau(A)+b\tau(B)\ \forall A,B\in\mathcal{A}, \forall a,b\in\mathbb{C}$$ The question rises, what are then the ...
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2answers
73 views

Derivative of the trace of $X^TP^TPX$ with respect to P

$\newcommand{\Tr}{\operatorname{Tr}}$ Consider the following expression: $\Tr(X^TP^TPX)$ where $X$ and $P$ are real matrices. What is the best way to approach the calculation of its derivative ...
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1answer
86 views

Characterize matrices A such that trace(AC)=0 for every matrix C with trace(C)=0

$A$ is an $n\times n$ matrix on the field $F$ such that for every $n\times n$ matrix $C$ with $\operatorname{trace}(C)=0$ we have $\operatorname{trace}(AC)=0$. Can we characterize such matrices $A $? ...
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1answer
218 views

Trace of symmetric positive semidefinite matrix when diagonalized (as a bilinear form) in a non-orthogonal basis

Let $\mathbf{S}$ be symmetric positive semidefinite matrix (i.e. one with all eigenvalues real and non-negative). Then there is an orthogonal matrix $\mathbf{U}$ (with its columns forming an ...
5
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1answer
186 views

solving equation also involving unknown matrix in trace

Given two real $m$ x $k$ matrices $A_1$ and $B_1$ and two $k$ x $k$ real matrices $A_2$ and $B_2$ I want to solve the following equation for $Q$. $Q$ is an orthogonal matrix, i.e. $Q^TQ=I$. ...
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1answer
77 views

Trace of power of stochastic matrix

I would like to know if this statement is true. Having a stochastic matrix (rows sum up to 1), with a positive (non-negative) diagonal, then it holds that $$\text{trace}({W^2})\leq ...
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1answer
55 views

Trace-zero functions in $W^{1,p}$

This is an excerpt of a textbook's proof for a theorem (Trace-zero functions in $W^{1,p}$), from PDE Evans, 2nd edition, page 275. Next let $\zeta \in C^\infty(\mathbb{R}_+)$ satisfy $$\zeta ...
2
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2answers
89 views

The trace of an operator

My question is derived from A. Deitmar's book: A First Course in Harmonic Analysis (second edition), p22, Exercise 1.17. Let me rewrite it again: Let $k:\mathbb{R}^2 \rightarrow \mathbb{C}$ be smooth ...
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1answer
59 views

Linear Algebra Question concerning the trace of a symmetric positive definite matrix.

The objective is to minimize the diagonal elements of a symmetric positive definite matrix. The expression of this matrix is a little bit nasty and its inverse is much easier to deal with. Can I claim ...
4
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1answer
71 views

Trace Theorem question

From PDE Evans, page 272. My question is towards the bootom of this post. THEOREM 1 (Trace Theorem). Assume $U$ is bounded and $\partial U$ is $C^1$. Then there exists a bounded linear operator ...