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

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1answer
21 views

Is $\langle A,B\rangle =\operatorname{trace}(AB^T)$ an inner product in $\mathbb R^{n\times m}$?

I don't understand why one should take transpose of $\operatorname{tr}(AB^T)$ and why we use the fact that $\operatorname{tr}(M)=\operatorname{tr}(M^T)$ for any $M$ that is a square matrix to solve ...
-1
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0answers
18 views

Which of the following are subspaces of $Mat_2(\mathbb{Q})$? [on hold]

Which of the following are subspaces of $Mat_2(\mathbb{Q})$? (i) The set of matrices which have zero trace. (ii) The set of matrices which have zero determinant. Do matrices refer to all matrices ...
2
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1answer
47 views

How can I solve for a , b , c , d?

Let's say I fix a list of two real numbers $\sigma = (\sigma_1, \sigma_2)$, and I want to show that there exists a real, entrywise-nonnegative matrix $A$ with $\sigma$ as its spectrum. How could I ...
0
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2answers
48 views

With these two equations, how do I show that either a,b,c,d must be negative, if v is not 0?

If I have the equations $$ad-bc = u^2 +v^2$$ $$a+d = 2u$$ and I want $a, b, c, d \ge 0$, then how I can show that this is impossible, if $v \ne 0$? I.e., if $v \ne 0$, then one of $a,b,c,d$ must ...
0
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1answer
38 views

Traces inequalities for projectors and reflections

Let $V$ - finite dimension space, $A : V \to V$ positive semidefinite operator, $P$ - orthoprojector ($P^2 = P = P^*$) $R$ - reflection ($R^2 = 1$, $R^* = R$). Can we say something nontrivial (any ...
0
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0answers
31 views

For which $A$ is true: $tr(XAY)=tr(YAX)$

$n \in \mathbb N,\forall X,Y \in \mathbb K^{n \times n},A \in \mathbb K^{n \times n} $ For which A is true: $tr(XAY)=tr(YAX)$ My answer would be if A is the identity matrix, but is there something ...
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0answers
29 views

When can $Tr(X^TAY)\geq 0$ when $A$ is positive semidefinite and $X,Y \neq 0$?

Under what conditions on $X,Y$ is $Tr(X^TAY)\geq 0$ when $A$ is square positive semidefinite but not the Identity matrix and $X,Y \neq \mathbb{0}$ (as in not equal to all matrix of all zeros)? $X,Y$ ...
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2answers
43 views

$n\cdot tr(AB)=tr(A) \cdot tr(B) $ $A$ is a scalar matrix

Let $A\in M^{n\times n}(\mathbb R)$. prove that if for every other $B\in M^{n\times n}(\mathbb R)$: $n\cdot tr(AB)=tr(A) \cdot tr(B) $, $A$ is a scalar matrix.
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1answer
30 views

Prove Or Disprove: $n\cdot tr(AB)=tr(A) \cdot tr(B) $iff $A$ or $B$ is scalar matrix [closed]

Prove/Disprove: $n\cdot tr(AB)=tr(A) \cdot tr(B) $ iff $A$ or $B$ is scalar matrix. A and B are square matrices of size n. So far I managed to prove that one side is right, left to right, that if A, ...
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0answers
8 views

Derivative of the singular values of the product of two matrices.

If $\mathbf{\Sigma}_{\mathbf{A}^H \mathbf{B}}$ is the diagonal matrix of singular values of $\mathbf{A}^H \mathbf{B}$, what is the derivative of ...
2
votes
1answer
56 views

Solving an equation involving the trace of a field

Let $F$ be a finite field of order $q$ where $q=2^{n}$ and fix $l\in F\setminus{0}$ with $Tr(l)=0$. I want to determine the number of $a$ such that $$Tr(la)=Tr(la^{-1})=1,$$ where $Tr$ denotes the ...
5
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1answer
83 views

Prove that trace of a matrix is $0$.

Let $ n\geq 2 $ and $ A,B,C \in M_{n}(\mathbb{C}) $ be three matrices so that $$ A^{2}B+BA^{2}=2ABA $$ and $ C=AB-BA $. Prove that $ \mbox{tr}(C^{k})=0,\forall k\in \mathbb{N}. $ I tried solving it ...
1
vote
1answer
39 views

How is the $H^{1/2}$ norm of function defined on a subset of the boundary?

Let $\Omega\subset \Omega^d$, $d\in \{2,3\}$, be a bounded $d$-polyhedron with $n$ faces. Denote the faces of $\partial\Omega$ as $\{e_i\}_{i=1}^n$. Let $u\in H^{1/2}(\partial\Omega)$ Taking the ...
10
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2answers
315 views

A conjecture about traces of projections

Let $M_n$ denote the space of all $n\times n$ complex matrices. Define $\tau:M_n\rightarrow \mathbb{C}$ by $$\tau(X)=\frac{1}{n}\sum_{i=1}^n x_{ii},$$ where of course $X=[x_{ij}]\in M_n$. Recall that ...
2
votes
1answer
52 views

Algebraic Number Theory,Marcus, Chapter 2, Question 16

In question 16 of chapter 2 in Marcus Book, I have to show that $\sqrt{3}\not\in\mathbb{Q}(\alpha)$,where $\alpha=\sqrt[4]{2}$ using the trace idea. the proof starts by assuming that ...
2
votes
1answer
47 views

Evaluate the mininal eigenvalue of positive definite matrix

Let $\{A_n\}$ be a sequence of positive definite matrices. Denote the minimal eigenvalue of matrix A by $\lambda_{min}(A)$. If $tr(A_n) \to +\infty$ when $n \to +\infty$, then what can we say about ...
2
votes
1answer
76 views

Proving existence of an element of trace 1

Let $F=\mathbb{F}_{q}$ be a finite field of order $q=2^{n}$ and let $\beta$ be a primitive element of $F$. I would like to prove that if $q>4$, then for each $1\leq i \leq \frac{q-2}{2}$, there ...
2
votes
2answers
51 views

Evaluate product of trace of two matrices

I have a question. Let A be a positive semi-definite matrix, H be a positive definite matrix. Is the following inequality right: $Tr(AH).Tr(AH^{-1}) \geq (Tr(A))^2$? I tried to take some concrete ...
1
vote
1answer
28 views

Understanding “trace of map” in the definition of harmonic maps

I have difficulty understanding "trace of map" in the definition of harmonic map. Let $\phi: (M,g)\to (N,h)$ is map between two Riemannian manifolds, the energy density is defined as ...
0
votes
1answer
19 views

Trace of Hermitian Positive Semidefinite Matrix

Well, the question I want to ask is as follows. Suppose A and B are Hermitian Positive Semidefinite (PSD) matrices, I wonder if it is possible to prove $Tr(A*(A+B)^{-1})\in (0,1]$ (if it is ...
0
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0answers
22 views

How to form a matrix using an endomorphism?

Let $f\in \mathrm{End}_\mathbb{R}(\mathbb{R}[x]_{<4})$ be the endomorphism given by $x^2 \dfrac{\mathrm{d}^2}{\mathrm{d}x^2} + \dfrac{\mathrm{d}}{\mathrm{d}x} + 2\mathrm{i}d$. I'm supposed to be ...
5
votes
2answers
105 views

$Tr(A^2)=Tr(A^3)=Tr(A^4)$ then find $Tr(A)$

Let $A$ be a non singular $n\times n$ matrix with all eigenvalues real and $$Tr(A^2)=Tr(A^3)=Tr(A^4).$$Find $Tr(A)$. I considered $2\times 2$ matrix $\begin{bmatrix}a&b\\c&d\end{bmatrix}$ ...
0
votes
2answers
29 views

Prove that $T_p$ is Self Adjoint

Let $P \in M^{\Bbb C}_{n \times n}$ an invertible matrix. Let $T_p$ be a linear transformation $T_p: M^{\Bbb C}_{n \times n} \to M^{\Bbb C}_{n \times n}$ such that: $$T_p(X)=P^{-1}XP$$. for every ...
1
vote
1answer
38 views

Prove that $\text{trace}(X^T Y)\le \sqrt{\text{trace}(X^T X)\text{trace}(Y^T Y)}$

As the title says, I would like help with proving that: $$ \text{trace}(X^T Y)\le \sqrt{\text{trace}(X^T X)\text{trace}(Y^T Y)} $$ where $X,Y\in\mathbb R^{n\times n}$. Perhaps this has to do with ...
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0answers
23 views

upper bound on trace

I have a $d \times n$ matrix $X$ consisting of $n$ vectors $x_i \in \mathbb{R}^d$. That is, $X = [x_1 x_2 \ldots x_{n-1} x_n]$. I also know that $\|x_i\|_2 \leq 1$. Let's say I have another matrix ...
0
votes
2answers
16 views

Derivative of trace of product of matrices

How to find the derivative of $$L=trace(A\left( \theta \right)^{-1}B\left( \theta \right))$$ wrt to $\theta \mathbf{}$. Where $A\left( \theta \right) $ and $B\left( \theta \right) $ are square ...
0
votes
1answer
33 views

Cauchy Schwarz inequality for trace

I found the following theorem in wikipedia. If matrices $A$ and $B$ are positive semi-definite of the same size then $trace(AB) \leq trace(A) trace(B)$. I need a reference book having this theorem to ...
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0answers
29 views

How to minimize the trace of a matrix

I've the following minimization problem: X * A > trace(X) Here X is a n x n matrix and A is a n x 1 matrix. The values in X are a set of real positive numbers R. I've tried to transform these ...
1
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1answer
39 views

Trace of a function on an elliptic curve

Let $K/F$ be a Galois extension of number fields with Galois group $G$. Let $E$ be an elliptic curve defined over $F$ and $f \in K(E)^{\times}$ be a function. Define the trace of $f$ to be ...
2
votes
1answer
59 views

What is the derivative of $\mathrm{trace}((S^T S)^{-2})$ w.r.t. $S$

I would like to compute the derivative of $\mathrm{trace}((S^T S)^{-2})$ w.r.t. $S$. I know that $\frac{\partial \mathrm{trace}((S^T S)^{-1})}{\partial S} = -2S(S^T S)^{-2}$ but I have a higher order ...
3
votes
1answer
45 views

Trace of multiplication operator on $L^2(\mathbb{T})$

Let $H=L^2(\mathbb{T})$, where $\mathbb{T}$ is the Torus. Consider a multiplication operator with a sufficiently nice function $f$. Is there somehow a formula like $$\mathrm{tr} M_f = C ...
2
votes
2answers
35 views

Similar matrices of rank 1

I am able to prove that two matrices of rank 1 are similar if and only if they have the same trace. But, my proof is long and complicated. In particular, I use the fact that the union of non-trivial ...
0
votes
1answer
55 views

Trace of matrix is sum of eigenvalues (positive semi-definite case)

Let $A \in \mathbb{R}^{n \times n}$. It is well-known that $\text{tr}(A)$ is equal to the sum of the eigenvalues of $A$. Let us know restrict $A$ to being positive semi-definite. Obviously, it is ...
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3answers
88 views

Trace of matrix that is a product of 2 others.

We consider that $A,B$ are two square matrices. I would like to know if there is a proof that $$tr(AB)=tr(BA)$$ I seek for special kind of proof without using sigma notation and matrices ...
2
votes
1answer
20 views

Is it true that for all matrices $A$ and all traceless matrices $T$, there exists a traceless matrix $T'$ such that $AT = T'A$?

Fix a real number $n$. By a "matrix", I mean an $n \times n$ real matrix. Now let $A$ denote a matrix. Is it true that for all traceless matrices $T$, there exists a traceless matrix $T'$ such that ...
0
votes
1answer
35 views

Find $\lim_a \frac{Tr^n \left( ({\bf x}-a\cdot{\bf y}) \cdot ({\bf x}-a \cdot{\bf y})^T \right)-Tr^n \left( {\bf x} \cdot {\bf x}^T \right)}{a}$

Let ${\bf x}, {\bf y} \in \mathbb{R}^{ m \times 1}$ and $a \in \mathbb{R}$. How to find the following limit \begin{align} \lim_{a \to 0 } \frac{Tr^n \left( ({\bf x}-a\cdot{\bf y}) \cdot ({\bf x}-a ...
2
votes
1answer
33 views

Trace of an algebraic number

Let $\alpha$ have minimal polynomial $m(x) = x^n + a_{n-1} x^{n-1} + \dots + a_0.$ Show that \begin{equation*} \mathrm{Tr}(\alpha^k) = -k a_{n-k} - \sum_{i=1}^{k-1} a_{n-i} ...
0
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0answers
10 views

the numbers Tr$(X^n)$ determine the conjugacy class of semisimple $X \in GL_n(\mathbb{C})$.

L.S., I am reading a paper of D. Blasius, where he states that the numbers Tr$(X^n) $ determine the conjugacy class of a semisimple element $X \in GL_n(\mathbb{C})$. I am having trouble proving this. ...
4
votes
1answer
118 views

Trace of the $k$-th Exterior Power of a Linear Operator

Let $V$ be an $n$ dimensional vector space over a field $F$ and $T$ be a linear operator over $V$. Assume that the characteristic of $F$ is not $2$. Definition. Consider the map $f_1:V^n\to ...
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votes
3answers
47 views

Dimension of trace-subspace

"Show that V, the set of all square matrices whose trace is 0, form a subspace of $M_{n,n}$ (the set of all square matrices). What is its dimension?" I have shown that $V$ is a subspace, but I do not ...
7
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1answer
64 views

Cayley-Hamilton Theorem - Trace of Exterior Power Form

Let $V$ be an $n$-dimensional vector space over a field $F$ (the characteristic of which, for the purpose of this post, may be taken as $0$). Let $T$ be a linear operator on $V$ and $\lambda\in F$. ...
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0answers
29 views

Derivative of trace of matrix product

Provided $\mathbf{P}$ is invertible, Solve \begin{equation} \frac{\partial}{\partial\mathbf{D}} ...
5
votes
2answers
267 views

Trace and the coefficients of the characteristic polynomial of a matrix

Let $A\in M(\mathbb F)_{n \times n}$ Prove that the trace of A is minus the coefficient of $\lambda ^{n-1}$ in the characteristic polynomial of A. I had several ideas to approach this problem - the ...
2
votes
3answers
76 views

Trace inequality on positive matrices

Let $A,B,C\geq 0$ be self-adjoint matrices. Assume $A\leq B$. Is it true that $$\mathrm{tr}(ACAC) \leq \mathrm{tr}(BCBC)?$$ How to prove this?
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vote
1answer
48 views

Trace operator and $W^{1,p}_0$

Let $W^{1,p}$ be the Sobolev space of $L^p$ functions with $L^p$ first derivatives. Let $W^{1,p}_0$ be the closure of the test functions in $W^{1,p}$. I am not explicitly writing the domain of the ...
2
votes
2answers
54 views

Find the inverse and determinant of A=(aI +T),

where is $a\ne 0$, $T$ has rank-one and zero trace. I just verified that a rank-one matrix has at most one non-zero eigenvalue. Now since T is of rank-one and has zero trace, that means all of its ...
3
votes
1answer
66 views

Show that $rank(A) \ngeq \frac{[tr(A)]^2 }{tr(A^2)}$

here $A$ is a Hermitian square matrix. A few thoughts on the problem: We know the matrix $A$ is unitarily diagonalizable, so $A = PDP^{-1}$ for some unitary matrix $P$. Since this is the only ...
0
votes
2answers
69 views

Rank = trace for idempotent nonsymmetric matrices

If $A$ is idempotent and symmetric, one can show that the rank of $A$ equals its trace. Is such equality preserved in general if we only know that $A$ is idempotent and not necessarily symmetric?
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0answers
62 views

How to Trace a Real-Life Flower Using Polar Equations?

Here is the flower I'm trying to trace: $\hskip2cm$ How can I trace this flower using polar equations? I currently have the formulas \begin{align} r_{1}&=1.75\sin(10\,\theta + 18) +3\\ ...
0
votes
0answers
22 views

Trace class with bounded operator

If I assume to have a trace class $A$ and a bounded operator $B$, how can I show that: $tr AB = \sum \langle \psi_n|AB| \psi_n \rangle < \infty$ I do understand by sense, that this should be, as ...