Questions tagged [trace]

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

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2
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1answer
57 views

Why does the trace show up in such expressions?

I've been studying different scattering processes (from Mandl & Shaw QFT's book, chapter 8) and there's always a purely-mathematical common step I do not understand: the showing-up of the trace. ...
0
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1answer
26 views

Computing a trace containing $\gamma$-matrices

I want to compute the following trace $$Tr \Big( Y(\not{\!p_1'}+m) \Big) \ \ (1)$$ Where $$\not{\!A} := \gamma^{\alpha} A_{\alpha} \ \ (2)$$ $$Y:= 4 \not{\!f_1} \not{\!p} \not{\!f_1} + m[-16(...
5
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0answers
51 views

Let $A\in\mathbb{R}^{2\times 2}$ and assume that $|{\rm tr}A|<4$. Prove or disprove that $b_1I_2+b_2A+b_3A^2+A^3=0$

Let $A\in\mathbb{R}^{2\times 2}$ and assume that $|{\rm tr}A|<4$. Prove or disprove that there exist $a_1,a_2,a_3,a_4\in\mathbb{C}$ such that $|a_1|<1$, $|a_2|<1$, $|a_3|<1$, $|a_4|<1$...
0
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1answer
35 views

For non-negative definite symmetric matrices, $\mathrm{tr}(AB)\le \mathrm{tr}(A)\mathrm{tr}(B)$

$\DeclareMathOperator{\tr}{\mathrm{tr}}$ Is the inequality in title true for non-negative definite matrices?? I could neither prove this result, nor provide a counter example. Context I was trying ...
0
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0answers
43 views

Maximization problem involving trace function

We have the following maximization problem $\max_X \text{Real}(\text{Tr} (AX)),~~~\text{s.t.}~\text{Tr}(X) \le P, ~~X \geq 0$ And $A$ and $X$ are complex square matrices of the same size. Is the ...
0
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0answers
24 views

Help with some gamma matrices trace identities

I need to understand the following derivation, but I can't understand which identities it used. Some help or more elaborate derivation will be great
1
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2answers
36 views

Find the determinant of matrix $A$

Let A be a 3 × 3 real matrix with zero diagonal entries. If $1 + i$ is an eigenvalue of A, the determinant of $A$ equals- I know the trace of the matrix is sum of eigen value but couldn't solve it.
1
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0answers
17 views

How can I use the tracelessness property of a matrix (H) to simplify the following commutation formula [H,A]?

I am trying to compute the matrix $F(0)$, where : $$F(t) = \frac{d}{dt}U A U^\dagger \tag{1}$$ where $$U(t) = e^{tH} \tag{2}$$ $H$ is a $N \times N$ anti-hermitian, traceless matrix and $A$ is a ...
0
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1answer
22 views

For which $C$ is the linear map $\varphi_C : V \to V, A \mapsto CA$ selfadjoint with scalar product $\langle A, B \rangle = \operatorname{tr}(A^t B)$?

Let $V=M_{n\times n}(\mathbb R)$. By the definition of selfadjoint, I have to show for which $C$, $\langle \varphi_C(A), B\rangle = \langle A,\varphi_C (B)\rangle$ is true $\forall A,B \in V$. In ...
2
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2answers
86 views

Who first invented/introduced the concept of the trace of a Matrix and Why?

Could anyone give any information about the invention of the concept of the trace of a Matrix, as this concept is so important and useful in linear algebra. I searched on the internet, but found ...
2
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3answers
63 views

Derivative of squared norm of component of a matrix perpendicular to identity matrix, with respect to the original matrix

Let $J\in\mathbb{R}^{n\times n}$ What is the derivative (with respect to $J$) of the squared norm of the component of $J$ that is orthogonal to $I$ (the identity matrix)? Attempt $J$'s projection ...
0
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1answer
29 views

Finding X from the matrix equation which includes trace function

Is there any way we can solve the following equation for $X$, where $X$, $A$, $B$ are square matrices of the same size and $\lambda$, $c$, and $a$ are scalar constants? $\lambda I = B + \dfrac{cA}{1+...
-1
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1answer
29 views

How to differentiate logarithm of trace function?

I want to find the derivative of $\log (1+\operatorname{Tr}(AX))$ w.r.t $X$. How can I do that?
-1
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2answers
49 views

Prove that $A = aI+B$ [closed]

Let $A \in M_{n\times n}(F)$. Show that $A = aI+B$, where $a \in F, B \in M_{n\times n}(F)$ and $\operatorname{tr}(B) = 0$. I just need a hint to start proving this. Solved
0
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3answers
50 views

If $A \in M_{2\times 2}(F)$ and $A^{2}=0$ then $tr(A)=0$

I actually saw an statement here on stackxchange: "Let $A$ be a square matrix $n \times n$. $trace (A)=0$, if and only if, $A^{2}=0$." So I could find a counterexample for →. And I think the ...
0
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2answers
40 views

Proving $\operatorname{Tr}((AB)^m)=\operatorname{Tr}((BA)^m)$

We know that $\operatorname{Tr}(AB)=\operatorname{Tr} (BA)$ How can we prove that $\operatorname{Tr}((AB)^m)=\operatorname{Tr}((BA)^m)$ ? I tried to use induction but it seemed that in the last step ...
6
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0answers
168 views

How to get the characteristic polynomial of this matrix?

Consider a $n\times n$ matrix: $$ M_n = \begin{pmatrix} a_1 & 1 & 0 & 0 & 0 & \cdots & 1 \\ 1 & a_2 & 1 & 0 & 0 & \cdots & 0 \\ 0 & 1 & a_3 &...
1
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1answer
66 views

How to prove that if $A$ is a square matrix on $\mathbb{R}$, $A$ is nilpotent, then trace($A$)=0

I need some help, I need to prove the following: Let $A$ be a square matrix on $\mathbb{R}$, if $A$ is nilpotent then Trace($A$)=0 I have seen some results for complex entries of the matrix, but what ...
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0answers
14 views

Lower bound on determinant of difference of matrices

Are there any lower bounds on the $\det(I+A^\top A - B^\top B)$? I'm looking for a bound that possibly depends on the 2-norm, Frobenius norm, or trace of $A$ and $B$. My attempt: Let $\lambda_i$ be ...
1
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1answer
59 views

How do you compute the partial trace in the oblique coordinates $a=u+\frac{1}{2}x, b=x$?

For my master's thesis I am reading the paper https://arxiv.org/abs/1204.5627 about quantum reference frame changes to center of mass coordinates. On Page 4, there is a calculation i just can't make ...
1
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1answer
18 views

Peierl Inequality for infinite-dim Hilbert space

Let $\mathscr{H}$ be a separable complex Hilbert space and $A$ be a self-adjoint bounded operator on $\mathscr{H}$. Let $h:\mathbb{R} \rightarrow [0,\infty)$ be convex. Then is it true that $$ \...
3
votes
1answer
36 views

Trace norm of rank one operator $x\otimes y$ for $x,y\in H$

Let $H$ be a Hilbert space. The trace norm on $B(H)$ is defined as $$\|u\|_{1}:=\operatorname{tr}(|u|):=\sum_{e\in E}\langle|u|(e),e\rangle,$$ where $|u|:=(u^{*}u)^{1/2}$ and $E$ is (any) orthonormal ...
1
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0answers
19 views

Trace and 2-norm of linear combination of outer products

Suppose that $c_i \in \mathbb{R}-\{0\}, B_i \in \mathbb{R}^{k \times m}, \alpha \in \mathbb{R}^k$ with $\|\alpha\|_2 = 1$. Consider the following linear combination of outer products: $$M = \sum_{i=1}...
1
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0answers
36 views

Mistake in Rasmussen & Williams 2006 or my mistake?

I wonder if I make any stupid mistake understanding this, or can there actually a mistake in the bible of Gaussian Processes: Rasmussen & Williams 2006: Gaussian Processes for Machine Learning? ...
2
votes
3answers
43 views

Notion of divisibility in a proof about number fields

I don't understand a point in the proof of the theorem 9.4 in Anthony Scholl's Number Fields lecture notes. Theorem: Let $K$ be a number field, $\mathcal{O}_K$ its ring of integers and $\alpha \in \...
0
votes
1answer
43 views

Variation of ln(det(A)) for diagonalizable and invertible matrix A

By using the following definition: $\delta ln(det(A))=ln(det(A+\delta A))-ln(det(A))$ I have to show that $\delta ln(det(A))=tr(A^{-1}\delta A))+O(\delta A^{2})$ My initial approach was to set ...
1
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0answers
41 views

If $A$ is an $n$-by-$n$ matrix with characteristic polynomial $(-1)^nt^n+b_{n-1}t^{n-1}+\cdots+b_1t+b_0$, show that $tr(A)=(-1)^{n-1}b_{n-1}$ [duplicate]

Let $A$ be an $nxn$ matrix with characteristic polynomial $$f(t)=(-1)^nt^n+b_{n-1}t^{n-1}+\cdots+b_1t+b_0$$ Show that $tr(A)=(-1)^{n-1}b_{n-1}$ I think I should be able to get this from the ...
1
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1answer
51 views

Spectral norm - trace inequality

I am wondering whether the following is true under which assumptions on A and B? $\operatorname{trace}(AB)\leqslant\|A\| \operatorname{trace}(B)$ The matrix norm is the spectral norm here. Maybe ...
0
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0answers
14 views

What is the difference between the trace operator and the contour integral operator?

I noticed that the trace operator can be used to trace the boundary of a subspace of a spectrum. Some methods also use the contour integral in spectral analysis, however I am not sure if this is ...
0
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0answers
13 views

Trace role in classification of SL(2,R) matrices.

I am working on J.Bochi and Avila's article "Uniformly hyperbolic finite-valued $SL(2,\mathbb{R})$-cocycles".As you may know,there is classification of $SL(2,\mathbb{R})$ which depends on the Trace of ...
1
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2answers
23 views

Question envolving trace and linear functionals.

I need to prove that $F^{n\times n} \approx (F^{n\times n})^*$ (dual space of $F^{n\times n}$) by isomorphism. My Attempt: Lets $f_A:F^{n\times n} \to F$, $f_A(X)= Tr(AX)$, it is obvious that $f \in ...
0
votes
1answer
19 views

What sort of matrix norms bound traces of products?

Suppose I have some linear operators $X_1, \dots, X_n$ on $\mathbb{C}^r$ (i.e. $r \times r$ matrices) and some other operators $Y_1^\epsilon, \dots, Y_n^\epsilon$ which are deformations of the $X_i$, ...
1
vote
1answer
32 views

Trace matrix calculation

The square of the Frobenius norm of an $n\times d$ matrix $X$ is equal to Tr($X^TX$). $$||X||_F^2 = \mathrm{Tr}(X^TX).$$ Then in the case of matrix involving addition or subtraction would be, for ...
-2
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1answer
35 views

Please I need help for this problem if ABA=0 then tr(AB)=0 [closed]

A, B are nxn matrices if ABA=0 then tr(AB)=0
4
votes
1answer
59 views

Modifying circulant Latin Squares

Question: Given a $N \times N$ circulant Latin square, $M$, is there a sequence of algorithmic modifications that one can make to $M$ such that the main diagonal will consist of exactly $2$ distinct ...
2
votes
1answer
47 views

If $(\lambda_i)$ are the eigenvalues of $A$, then $\sum_{i=1}^k\lambda_i=\sup_{\text{rank}B=k}\langle AB,B\rangle_{HS}$

Let $H$ be a $\mathbb R$-Hilbert space, $N:=\mathbb N\cap[0,\dim H]$, $A\in\mathfrak L(H)$ be compact and self-adjoint and $I:=\mathbb N\cap[0,\operatorname{rank}A]$. By the spectral theorem, $$A=\...
-1
votes
1answer
21 views

How to show $ tr(AB^\intercal)=\sum\limits_{i=1}^n\sum\limits_{j=1}^n a_{ij}b_{ij}$

Show that for any two matrixes $A = 􏰀a_{ij}$ with $1\leq􏰁i,j\leq n , B = 􏰀b_{ij}$ with $1\leq􏰁i,j\leq n$. $tr(AB^\intercal)=\sum\limits_{i=1}^n\sum\limits_{j=1}^n a_{ij}b_{ij}$ I'm confused as ...
1
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1answer
76 views

Trace Theorem: a question about Evans' proof

This is a part of the proof of the Thoerem "Trace-zero functions in $W^{1,p}(\Omega)$ in the book of Evans. I don't understand the inequality involving $\displaystyle\int_{\mathbb{R^N}_{+}}\vert Dw_m -...
2
votes
0answers
36 views

Linear Algebra, Matrix and Trace.

I need to bound the trace of a matrix. Can anyone help me see where the following bound comes from, I have spent ages trying to do the calculation but cant see it ! I will provide my working so far :...
0
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0answers
11 views

Can Peiels-Bogoliubov inequality be extended to infinite-dimensions?

I am interested in trace inequalities and especially the Peierls-Bogoliubov inequality. Wiki has some information and references on the subject, but as far as I've checked, the proofs are restricted ...
1
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2answers
59 views

Trace of a Matrix Product.

Let $A \in \mathbb{R}^{n\times m}$, and $B\in \mathbb{R}^{m\times m}$. Let $A'$ denote the transpose of $A$. From this we know that : $A'\in \mathbb{R}^{m\times n}$ and $AB\in \mathbb{R}^{n\times ...
0
votes
1answer
16 views

Can a sequence of operators with trace-class norm 1 have a trace that converges absolutely to 0?

Does there exist a sequence of (non-normal) trace-class operators $X_n$ such that in some ONB $\{ v_i \}_{i \in \mathbb{N}}$ we have \begin{align*} \sum_{i=1}^\infty \vert \langle v_i, X_n v_i \...
0
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0answers
52 views

Is the trace of the matrix exponential convex or non-convex?

I am trying to understand whether the following expression is a convex function: $$ f\left (\mathbf{X} \right ) = \mathrm{tr}\left( \left ( \mathbf{\Lambda}+\alpha_0 \mathbf{\Psi}^T \left( I-e^\...
0
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1answer
66 views

Taylor-like bound for the matrix exponential tr(exp(X))

I am trying to prove the following Taylor-like bound for the the matrix function $F(X) := \operatorname{tr}(\exp(X))$, and symmetric matrices $X, V$ with $\lVert V \rVert \leq 1/2$: $$ F(X+V) \leq F(X)...
1
vote
1answer
50 views

Prove that $\operatorname{trace}(A^{-1} \frac{dA}{dt}) = \sum_i^n \frac{\lambda_i'}{\lambda_i}$.

When proving Jacobi's formula for an invertible differentiable matrix $A(t)$ since $$ \det A(t) = \prod_i^n \lambda_i(t) $$ where $\lambda_i$ are the generalized eigenvalues, we get $$ \begin{...
0
votes
2answers
32 views

Derivatives of trace of complicated function

When $A$, $B$ and $C$ are positive semi-definite and hermitian symmetric matrices, what is the derivative and second derivatives of $f(x) = tr(x^2 A (xB + C)^{-1})$, for $x>0$ ?
0
votes
2answers
42 views

Why sometimes the trace of a inverse matrix is very high and sometimes is low?

I have an implementation that produces a matrix $A$. I am working on the trace of it's inverse $tr(A^{-1})$. In my problem, there are just two states for the result. Sometimes the result is very high ...
1
vote
1answer
30 views

Relate l1 norm of Hadamard product to trace

Suppose $A$ is a $q\times p$ matrix, $B$ is $q \times p$ matrix, $A_j$ is the jth column of $A$, and $B_j$ is the jth column of $B$. The following sum of $l_1$ norms, where "$\circ$" is the ...
2
votes
1answer
42 views

Trace Form on Product of Fields

I am studying algebraic number theory and I am having trouble understanding something. Let $K$ be a number field with ring of integers $\mathcal{O}_K$. Suppose a prime $p$ does not ramify in $K$. Then ...
2
votes
2answers
495 views

A linear transformation such that $T(AB)=T(BA)$

The question goes as follows: Let $V$ be a vector space and let $T: M_{2 \times 2} (R) —> V$ such that $T(AB)=T(BA)$ for all $A, B \in M_{2 \times 2}$. Show that $T(A) = 1/2(trA)T(I2)$ for all $A \...

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