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

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-1
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0answers
13 views

inequalities for $tr(AB)$ , where A and B symmetry, positively definite matrix

Let $A$ and $B$ be two symmetry, positively definite $n\times n$ matrix with positive eigenvalue $a_1,...,a_n$ and $b_1,...,b_n$ respectively. What's the relationship between them and $tr(AB)$? Are ...
1
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0answers
44 views

Integration.Matrix.Determinant.Inverse.Trace.

Given $$ I_n=\int_0^1\frac{x^n}{x^{2012}-1}{\rm d}x\text{ and }J_n=\int_0^1\frac{x^n}{x^{2013}+1}{\rm d}x\quad\forall n>2012, n\in\mathbb N$$ If the matrix $$\rm A=[a_{ij}]_{3\times3}\text{ where ...
2
votes
0answers
41 views

algebraic equation with trace

I have a problem which I don't know how to attack. Actually I am not even sure there is a way to do it. Is it possible to solve an equation of this form $$A²-\frac{tr(A)²}{4}Id_{4\times 4}=B$$ where ...
0
votes
1answer
28 views

Does $\lim_{N\rightarrow\infty}\frac{tr(A'A)}{N}=0$ imply $\lim_{N\rightarrow\infty}\frac{tr(A)}{N}=0$?

$A$ is an $N\times N$ matrix with bounded row and column norms. Does $\lim_{N\rightarrow\infty}\frac{tr(A'A)}{N}=0$ imply $\lim_{N\rightarrow\infty}\frac{tr(A)}{N}=0$? I know this is true for ...
0
votes
2answers
86 views

How to prove that tr(A) = tr(B) given that B is similar to A [duplicate]

If A and B are similar, how does one prove that tr(A) = tr(B)
0
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0answers
22 views

Trace minimization when some matrix is unknown

The problem is as follows: $\displaystyle\min_{V}$ trace($V^TH^T\Phi HV$)$\\$ s.t. $V^TV=I_d$ in the case when $H$ is not known. When $H$ is known, the solution is given by the eigenvectors ...
1
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0answers
36 views

Let $A,B$ be two $3\times 3$ matrices with complex entries, such that $BA^2=A^2B$. Prove that $\det(AB-BA)=0$

Problem: Let $A,B$ be two $3\times 3$ matrices with complex entries, such that $BA^2=A^2B$. Prove that $\det(AB-BA)=0$ attempt: $A(AB-BA)=A^2B-ABA=BA^2-ABA=(BA-AB)A=-(AB-BA)A$ So ...
1
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1answer
30 views

Can find the determinant of a matrix A of size $n$ in terms of the traces of $A^m$ [duplicate]

We can find the determinant of a matrix A of size $n$ in terms of the traces of $A^m$, for $m=1,…,n$ ? It's det of a matrix with term are traces, but i saw but i can't remember Expressing the ...
1
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2answers
31 views

Show the Trace$(B)^{2} \leq$ nTrace$(B^{T}B)$

The following definition is needed for my actual question: For $A, B \in \mathcal{M}_{n \times n}$ define $$ \langle A, B \rangle = \text{Trace}(B^{T}A) = \sum_{j=1}^{n}\sum_{i=1}^{n}b_{ij} \, ...
-1
votes
2answers
56 views

If $\operatorname{trace}(Z)=0$, then there exist $X$, $Y$, $|Y|\not=0$ satisfy $Z=XY-YX$ [closed]

Lemma: if $\operatorname{tr}(Z)=0$,then there exist $X$, $Y$, $|Y|\not=0$ satisfy $Z=XY-YX$. It's an old problem but I can't prove it.
0
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2answers
25 views

Does the trace and determinant uniquely determine the eigenvalues of a 3 by 3 matrix with algebraic multiplicity of 2?

I have a 3 by 3 matrix $M$ whose eigenvalues are $a$, $b$, and $b$. The determinant and trace of $M$ are known from its eigenvalues: $det(M)=ab^2$ and $Tr(M)=2b+a$. I wanted to show that if ...
0
votes
1answer
15 views

Prove that $\nabla_X tr(X^TB)= B $ where $B \in \mathbb{C^{m*n}}$ and $X \in \mathbb{R^{m*n}} $

Prove that $\nabla_X tr(X^TB)= B $ where $B \in \mathbb{C^{m*n}}$ and $X \in \mathbb{R^{m*n}} $ and $\nabla_X$ is the derivative with respect to X. How can I prove the above?
0
votes
2answers
71 views

The trace of a symmetric matrix

If A is an $n\times n$ symmetric matrix with eigenvalues $c_1 \ge ... \ge c_n$, and $U$ is an $ n \times k$ semi-orthogonal matrix, with $ k \le n$, how to prove that $\text{tr}(U^TAU) \le ...
0
votes
0answers
11 views

Quadratic form expressed with trace.

I am attempting to prove the following identity: $(x-a)A^T(x-a)=\text{tr}(Ax_cx_c^T)+n(a-\bar{x})^2 \text{tr}(A)$ where $x_c=(x-\bar{x})$ and the orders of the vectors are $n$. I got as far as: ...
3
votes
1answer
35 views

Is the trace of a matrix a norm?

If the matrix norm of A is defined as $\|A\|=\sum_{i,j}|Aij|$ then how do I determine if the sum of the diagonal elements, i.e., the trace is a valid norm? I am not really sure how to approach this ...
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votes
0answers
20 views

prove $A$ is nilpotent if $tr(A^k)=0$ for all $k>0$ [duplicate]

I am beginner in linear algebra. how can I prove if $tr(A^n) = 0$ for all n>0 then we can say A is nilpotent.
0
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1answer
21 views

Trace and Spectral norm of a matrix

Let $A_{n\times n}$ be a matrix. How I can show $$\vert \operatorname{trace} (A) \vert \leqslant n \sqrt{\rho(A^T A)}= n \Vert A \Vert_2$$ and if $A$ is symmetric and positive definite, ...
3
votes
1answer
74 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 ...
1
vote
1answer
63 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: ...
0
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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
18 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
34 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. ...
3
votes
0answers
126 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
votes
0answers
31 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 $\gamma \in GF(q^m)$ ...
0
votes
0answers
22 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 ...
2
votes
3answers
62 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 ...
0
votes
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} ...
1
vote
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
20 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
votes
1answer
33 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
votes
0answers
27 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
17 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 ...
0
votes
1answer
46 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 ...
1
vote
2answers
97 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) ...
1
vote
1answer
71 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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votes
1answer
12 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 $|Tr[A]|\geq |Tr[B]|$?
2
votes
1answer
116 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 ...
0
votes
1answer
36 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. ...
0
votes
1answer
41 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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vote
2answers
50 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 ...
1
vote
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: ...
0
votes
1answer
39 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
votes
1answer
30 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 ...
3
votes
1answer
55 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 ...
1
vote
1answer
34 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 ...
0
votes
0answers
25 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 ...
1
vote
0answers
22 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 $K_p$ ...
2
votes
0answers
51 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 ...
-2
votes
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 ...
0
votes
2answers
54 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 ...