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

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17 views

Maximize trace of matrix equation given two constraints

Let $\mathbf{Q}$ be a rotation matrix and $\mathbf{A}$ and $\mathbf{B}$ be two real-valued matrices of the same size. I want to maximize the function $$ f(\mathbf{Q})=tr\;\mathbf{QA} \qquad ...
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1answer
32 views

trace inequalities: linear algebra

If S is any $n \times n$ real, symmetric, invertible matrix and D is any $n \times n$ diagonal matrix such that $0\prec D \prec I$ then does there exist a constant $\gamma$ such that: ...
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0answers
13 views

Is $\langle A,B \rangle = \text{trace}(A^{T}B)$ undefined in $\mathbb{R}^{n\times 1}$ and $\mathbb{R}^{1\times m}$?

$A^TB$ would be a $n\times 1$ or $1\times m$ vector in each case, no? How can we sum diagonal elements if they don't exist?
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0answers
13 views

The differential of the trace of two random matrices

I have two random matrices evolving in time, $X_{t}$ and $Y_{t}$. I know that $dX_{t} = X_{t}Adt + X_{t}dB_{t}$ and $dY_{t} = AY_{t}dt + Y_{t}dB_{t}$, where $A$ is a constant matrix and $dB_{t}$ is ...
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0answers
31 views

Partial Trace of Density Operator

Before stating my question I present my motivation: to learn more about the tensor product. Now, quantum mechanics assigns a Hilbert space to each physical system as a postulate of the theory. ...
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0answers
124 views

Why the equality of spectral zeta functions imply the isospectrality?

Let $\Delta_{M_1}$ and $\Delta_{M_2}$ be the Laplace-Beltrami operators on two compact and connected Riemannian manifolds $M_1$ and $M_2$ respectively. We define the spectral zeta function (or ...
3
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0answers
30 views

Finite field question involving the trace and a permutation.

Let $q$ be a power of a prime $p$, and $m,l$ positive integers with gcd$(l,q^m-1)=1$. Denote $Tr$ to be the trace of $GF(q^m)$ over $GF(q)$. Suppose that there exists a nonzero $\gamma \in GF(q^m)$ ...
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19 views

Trace of Matrices multiplication

Problem: I want to obtain the trace: $tr\left( {{A^H}{C^{ - 1}}A} \right)$ where $A$ is a $S \times L$ matrix, $A^H$ is conjugate transpose of $A$, $C$ is a $S \times S$ matrix, ${A^H} A=I$ identity ...
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3answers
59 views

If $A \in M(n , \mathbb R) $ be such that $A^2=A$ , then is it true that $rank (A)=Trace(A)$ ?

If $A \in M(n , \mathbb R) $ be such that $A^2=A$ , then is it true that $rank (A)=Trace(A)$ ? What I have done is to just observe that all the eigenvalues must be either $1$ or $0$ ; but no headway ...
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1answer
35 views

Ortogonal matrix simple identity?

In order to prove the invariance of the trace of a tensor under the transformation $\tilde{T}^{i,j}=\Sigma_{k,l} O^i_kO^j_lT^{k,l}$ where $O\in SO(3)$ I have to prove that $\Sigma_{k,l} ...
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0answers
42 views

Trace in finite Fields

(1) In $\mathbb{F}_{2^n}$ with odd n it should be shown that half of the Trace values are 0 and the other part is 1 with help of the additivity of the Trace. (2) Now n shall be even. Now the ...
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1answer
19 views

Factoring out the trace of a matrix

This question is related to a derivation step in " A Duality View of Spectral Methods for Dimensionality Reduction" Xiao et al. 2006 When deriving the dual equation for Maximum Variance Unfolding ...
2
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1answer
32 views

Bijections of a finite field that preserve the kernel of the trace

Let $q=p^n$, for some prime $p$ and positive integer $n$. Let $m$ also be a positive integer, and denote $Tr$ to be the trace of $GF(q^m)$ over $GF(q)$. I have the suspicion that all the functions ...
2
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0answers
26 views

Inclusion of subring in Ideal

Let $K$ be a commutative ring with mutpilicative identity and $m \ge 3$. Let $L(m,K)$ be a subring of Lie ring of matrices with coefficients from $K$ and traces = $0$: $ \{ (a_{ij}) \in M_m (K) | ...
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1answer
10 views

Is it possible to express this term by using “trace function”?

I wonder if we can express the following term by using "Trace function"? $$(X-\mu)^T \Sigma^{-1}(X-\mu)$$ This is the quadratic term in Multivariate Gaussian Distribution with mean of $\mu$ and ...
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1answer
44 views

Let $\rho : G \rightarrow GL_n(\mathbb{C})$ be a representation show that $|\operatorname{tr} X| \leq \dim \rho$

Let $G$ be a finite group. Let $\rho : G \rightarrow GL_n(\mathbb{C})$ be a representation, pick $g \in G$ and write $X=\rho(g)$. Prove that all eigenvalues of $X$ are roots of unity, and deduce that ...
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2answers
93 views

Prove that $\det(A^p+B^p)=\det (A^p)+\det(B^p) +\operatorname{tr}\left(\left(A\operatorname{adj}(B)\right)^p\right)$

Let $A,B$ be $2\times 2$ matrices such that $AB=BA$. Prove that for every positive integer $p$: $$ \det(A^p+B^p)=\det (A^p)+\det(B^p) ...
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1answer
61 views

Derivative of trace of inverse matrix?

I've been trying to derive the formula for the derivative of $Tr(X^{-1})$ w.r.t. $X$, which I know is $X^{-2T}$. According to the Matrix Cookbook $$\dfrac{\partial g(U)}{\partial X_{ij}} = ...
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1answer
11 views

$A^{\dagger}A\geq B^{\dagger}B$. Can we say that $|Tr[A]|\geq |Tr[B]|$?

Suppose that A and B are any two square matrices of equal dimension with complex entries and $A^{\dagger}A\geq B^{\dagger}B$. Can we say that $|Tr[A]|\geq |Tr[B]|$?
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1answer
108 views

If $Tr(A)=0$ then $T=R^{-1}AR$ has all entries on its main diagonal equal to $0$

Prove that if $A$ is a square matrix and $Tr(A)=0$, then there exixts an invertible matrix $R$ such that the matrix $T=R^{-1}AR$ has all entries on its main diagonal equal to $0$. It seems like ...
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1answer
35 views

Prove that $Tr((AB^2)A)=Tr(A^2B^2)$

Prove that for every $n\times n$ matrices $A,B$: $$Tr((AB^2)A)=Tr(A^2B^2)$$ I need a solution that doesn't use expansion. One more question comes into my mind: given $A,B$ are square matrices. ...
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1answer
34 views

Norm and Trace of an element is an integer, then element is an integral?

Let $L/K$ be a finite field extension, and let $\{b_1,b_2,...,b_d\}$ be a basis for $L/K$ My notes define $O_k:=\mathbb{B}\cap K$, where $\mathbb{B}:=\{\alpha$ is algebraic|min poly of $\alpha$ ...
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2answers
49 views

Calculate the trace of all elements in $F_8$

I got the following exercise where you have to calc the trace of all elements in ${F_8}$ which is constructed as ${F_2}[x]$/(${x^3+x+1}$)${F_2}[x]$. Up to now I did those steps: 1) Find all elements ...
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1answer
24 views

Eigenvalus and trace of matrix

Given matrix $A = \begin{bmatrix}1 & -2 & -1 \\ 1 & 2 & 1 \\ -1 & 0 & 1\end{bmatrix}$. I have to find its eigenvalues and trace of the matrix $A^{2014}$. I found engenvalues: ...
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1answer
36 views

Remove the Kronecker operator in $\mathrm{trace((\Sigma^{-1}\otimes S^{-1})ZDZ^{T}})$

I am not sure if I can remove the Kronecker operator in the following formula $$\mathrm{trace((\Sigma^{-1}\otimes S^{-1})ZDZ^{T}}),$$ where $\Sigma,S, D$ are all positive-semidefinite and symmetric. ...
2
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1answer
26 views

Is a bounded operator with finite trace trace class?

Let $\mathcal{H}$ be a seperable Hilbert space, $A\in\mathcal{B}(\mathcal{H})$ a bounded linear Operator and assume we have an orthonormal basis $(x_n)_{n=1}^\infty$. If $A$ is trace-class, then ...
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1answer
53 views

I have a problem with finding the trace to a matrix.

Let $T$ be a matrix of which I know its characteristic values, how can I find $\operatorname{Tr}(T-I)^{-1}$? I know that the sum of the characteristic values is the trace, but I'm having a problem as ...
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1answer
28 views

Question of $u\in L^p(U)$ does not have a trace on $\partial U$. [duplicate]

Let $U$ be bounded, with $C^1$ boundary. Show that a "typical" function $u\in L^p(U)$ does not have a trace on $\partial U$. More precisely, prove that does not exist a bounded linear operator ...
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0answers
24 views

convexity of inverse of a matrix

I know that the function $f(X)$ which maps matrix $X$ to $Tr((X)^{-1})$ is convex for symmetric positive definite $X$. This has also been answered in Is the trace of inverse matrix convex? for ...
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0answers
21 views

Unit balls and the Schatten norms

I have a very naive question: Let $A$ and $B$ $n \times n$ (complex) matrices with operator norms $\|A\| \leq 1$ and $\|B\| \leq 1.$ Pick a $1 \leq p < \infty.$ Then with a constant $K_p$ ...
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0answers
50 views

Is it possible to prove that $\text{tr}(AB)=\text{tr}(BA)$ without using matrices? [duplicate]

Is it possible to prove that $\text{tr}(AB)=\text{tr}(BA)$ without using matrices or having to choose a particular base ? Such a proof should probably use a non matricial definition of traces. One ...
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1answer
26 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 ...
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2answers
53 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
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1answer
30 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
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1answer
44 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 ...
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15 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
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2answers
90 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
55 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 ...
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1answer
42 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 ...
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0answers
20 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 ...
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6answers
115 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 ?
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0answers
48 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
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1answer
100 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
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2answers
37 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: ...
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3answers
33 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 ...
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3answers
71 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 ...
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2answers
39 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) $
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1answer
49 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
44 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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3answers
54 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 ...