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

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2
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3answers
50 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
30 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
39 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 ...
1
vote
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
votes
1answer
28 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
25 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) | ...
0
votes
0answers
4 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
43 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
91 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
49 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}} = ...
-1
votes
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 $|Tr[A]|\geq |Tr[B]|$?
2
votes
1answer
96 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
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. ...
0
votes
1answer
32 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 $\alpha$ ...
1
vote
2answers
44 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
35 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
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 ...
3
votes
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 ...
1
vote
1answer
27 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
22 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
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 $K_p$ ...
2
votes
0answers
47 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
51 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
29 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
42 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
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
votes
2answers
87 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
votes
0answers
51 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
votes
1answer
39 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
vote
0answers
18 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
108 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
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
votes
1answer
91 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
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: ...
0
votes
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 ...
1
vote
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 ...
0
votes
2answers
38 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
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 ...
0
votes
1answer
43 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
49 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
vote
2answers
90 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
votes
0answers
30 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
21 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
19 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
22 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 ...