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

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-1
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1answer
16 views

Is the finding trace of the Riemann tensor the same thing as contracting two indices?

To form the Ricci curvature tensor, we have to take the trace of the Riemann tensor. But I also know \begin{equation} R_{ij} := R_{kij}^{\phantom{kij}k} \end{equation} Can someone show me why ...
0
votes
2answers
46 views

Trace of a $2 \times 2$ matrix

Let $\mathrm{A}$ a $2 \times 2$ matrix such that $\mathrm{I}\neq\mathrm{A}\neq\mathrm{-I}$, where $\mathrm{I}$ is the $2 \times 2$ identity matrix. If $\mathrm{A}=\mathrm{A}^{-1}$, find the trace of ...
2
votes
1answer
18 views

Are trace function embedded in $L^\infty$?

Consider a bounded domain $\Omega \subset \mathbb R^d$ with a Lipschitz boundary (could also be a smooth boundary). Is the trace space $H^{1/2}(\partial\Omega)$ embedded in $L^\infty(\partial\Omega)$? ...
3
votes
1answer
41 views

Geometrical or Physical significance (interpretation) of the inner-product $\langle A,B \rangle := Trace (AB^t)$ over $M_n(\mathbb R)$

$\langle A,B \rangle := Trace (AB^t)$ is an inner product over the vector space $M_n(\mathbb R)$ of all real matrices of size $n$ , I would like to know whether this inner-product has any Geometrical ...
1
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0answers
14 views

Summation of elements of a matrix in matrix notation

I have come across the following proof in a research paper. I feel the given end formula is wrong. Any help to correct it is greatly appreciated. where $V = \{v_{ij}\} \hspace{0.3cm} \text{with} ...
3
votes
2answers
84 views

For which values of $a$ the matrix is diagonalizable

Given the following matrix: $$B=\begin{bmatrix} 1 & 0 & 0 \\ 1 & 0 & a^2 \\ 1 & 1 & 0 \end{bmatrix}$$ I tried to find for which values of $a$, the matrix $B$ is diagonalizable. ...
3
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0answers
44 views

quadratic form from nxn matrices to reals ( Tr(A^2) ). I need to find it's signature and rank.

Firstly prove $Tr(A^2)$ defines a quadratic form from the space of $n \times n$ matrices to R. I think you just have to show that $Tr(A B)$ is a bilinear form which seems too easy to be correct or I'm ...
0
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1answer
32 views

The Trace Theorem for $W^{1,p}$ functions

I'm trying to understand the proof of the trace theorem. This is from a course I am taking, so I will write out what we have done explicitly. $\textbf{Trace Theorem}$ Suppose $\Omega ...
1
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0answers
15 views

Proof of existence of trace map $T:H^1(\mathbb{R}^n_+) \to H^{\frac 12}(\mathbb{R}^{n-1})$ not using the Fourier transform

I'm looking for a proof of the existence of the trace map $T:H^1(\mathbb{R}^n_+) \to H^{\frac 12}(\mathbb{R}^{n-1})$ which does not use the Fourier transform. In particular, I want to prove the ...
1
vote
6answers
103 views

Prove that $\operatorname{Trace}(A^2) \le 0$

Let $A \in M_n(\mathbb{R})$ is a antisymmetric matrix such as $A^T=-A$. Prove that $\operatorname{Trace}(A^2) \le 0 $ I see that, for some matrix such as, their terms in diagonal are negative ?
0
votes
0answers
46 views

How to interpret some matrix lemmas on Wikipedia - the number 1 vs. the matrix I

I'm reading some lemmas on Wikipedia, eg, the Matrix determinant lemma, and the Sherman-Morrison formula, and both of these formulas have a 1 added to a product of column vectors and matrices. How ...
3
votes
1answer
71 views

Questions about matrix rank, trace, and invertibility,

(a) Prove that a square matrix $T$ of rank one has $\text{tr}(T)=0$ if and only if $T^2=0$. (b) Consider a matrix $A$ of the form $A=aI+T$, where $a\ne0$, $I$ is the identity matrix, and $T$ has ...
3
votes
2answers
35 views

Upper bound on trace of product of unitary and arbitrary matrix

Let the field be complex, $U$ be an $n\times n$ unitary matrix, $M$ be any $n\times n$ matrix, and $|M|$ denote the matrix formed by taking the absolute value of every entry of $M$. Edited Question: ...
0
votes
3answers
31 views

If $T : F^{2 \times 2} \to F^{2\times 2}$ is $T(A) = PA$ for some fixed $2 \times 2$ matrix $P$, why is $\operatorname{tr} T = 2\operatorname{tr} P$?

I am asked to prove that if $T$ is a linear operator on the space of $2 \times 2$ matrices over a field $F$ such that $T(A) = PA$ for some fixed $2 \times 2$ matrix $P$, then ...
1
vote
3answers
69 views

Prove that if $\mathbf A$ is an $n\times m$ matrix, then $\text{tr}(\mathbf A \mathbf A')=\text{tr}(\mathbf A' \mathbf A) $

If $\mathbf A$ is an $n\times m$ matrix, then $\text{tr}(\mathbf A \mathbf A')=\text{tr}(\mathbf A' \mathbf A) \text{ where } \mathbf A'\text{ is transpose of }\mathbf A\text{ and tr}(\mathbf A ...
0
votes
1answer
31 views

Proving $\mathbf x$ is a $n\times 1$ vector and $\mathbf A$ an $n\times n$ matrix, then $\mathbf x'\mathbf A \mathbf x = \text{tr} (\mathbf {Axx}')$

If $\mathbf x$ is a $n\times 1$ vector and $\mathbf A$ an $n\times n$ matrix, then $\mathbf x'\mathbf A \mathbf x = \text{tr} (\mathbf {Axx}') (\mathbf A'=transpose A) $
0
votes
1answer
48 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 ...
0
votes
1answer
41 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 ...
1
vote
3answers
43 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 ...
1
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2answers
84 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
votes
1answer
63 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 $tr(\mathbf{A} + \mathbf{B} ) = tr(\mathbf{A}) + tr(\mathbf{B})$ ...
0
votes
0answers
14 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 ...
0
votes
1answer
16 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 ...
0
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0answers
29 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 ...
0
votes
0answers
20 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 ...
0
votes
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 $$ ...
1
vote
1answer
19 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 ...
0
votes
0answers
14 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
votes
2answers
47 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 ...
2
votes
1answer
42 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 ...
1
vote
1answer
24 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 ...
0
votes
1answer
30 views

Trace of logarithm 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
votes
0answers
34 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 ...
0
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1answer
35 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 ...
1
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1answer
40 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 ...
1
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1answer
26 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
votes
1answer
74 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 ...
1
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1answer
35 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 ...
2
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0answers
30 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
42 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)
2
votes
2answers
50 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 ...
0
votes
3answers
54 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 ...
0
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1answer
48 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}} ...
0
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1answer
47 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 ...
1
vote
1answer
71 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 ...
1
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2answers
87 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 ...
0
votes
3answers
184 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.
1
vote
1answer
42 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 ...
1
vote
0answers
20 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 ...
0
votes
1answer
56 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 ...