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Questions tagged [trace]

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

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Assume that for a n × n-matrix A with real matrix elements we have A$^2$ = −E. Prove that tr(A) = $0$.

Assume that for a n × n-matrix A with real matrix elements we have A$^2$ = −E. Prove that tr(A) = $0$. Any ideas ?
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1answer
53 views

Find $Tr(B)+Tr(C)$

If $B,C$ are $2 \times 2 $ matrices with integer entries such that: $$\begin{bmatrix} -1 &1 \\ 0& 2 \end{bmatrix}=B^3+C^3$$ Find value of $Tr(B)+Tr(C)$ My try: Taking trace on both sides ...
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1answer
62 views

Non-trivial kernel

Am I correct in saying that this is a group homomorphism? If this is a group homomorphism does it have a non-trivial kernel? $$\Phi : (M(\mathbb{R},n), +) \longrightarrow (\mathbb{R}, +) : A \mapsto ...
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1answer
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Trace and Frobenius Norm

For a matrix $A \in \mathbb R^{n \times n}$, prove that $|tr(A)|\leq \sqrt{n}||A||_{F}$, where $tr(A)$ denotes the trace of the matrix A, and $||.||_{F}$ denotes the Frobenius norm. We know that $|tr(...
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0answers
48 views

How do I calculate $e^{tA}$

I want to calculate $e^{tA}$, and eigenvalues are $\lambda_1=$ trace A , $\lambda_2 = 0 \DeclareMathOperator{\tr}{tr}$ so $P_0=I$ and $P_1 =(A-\lambda_1I)=A-(\tr A)I$ $r_1=e^{(\lambda_1)t} = e^{(\tr A)...
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1answer
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How do i generalize theory to arbitrary trace and arbitrary determinant?

Given a matrix $A\in \Bbb R^{2\times 2}$ Assume that trace $A = 0$. Then: a. If $\det A = 0$, then $0$ is the only eigenvalue. b. If $\det A <0$, then eigenvalue is $\pm\sqrt{-\det A}$ c. If $\det ...
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1answer
38 views

What is the derivative of $\operatorname{trace}(XCP(XC)^T)$?

I am really stuck at calculating $\frac{d\operatorname{trace}(XCP(XC)^T)}{dC}$ where $P \in R^{r\times r}$, $X \in R^{m\times n}$ and $C \in R^{n\times r}$ . Do I need to recall $A=XC$ and then apply ...
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1answer
36 views

Non-diagonalizable compact operators and the trace-class condition

For a compact operator $A$ on a Hilbert space, it is said that $A$ is trace-class if for some (and hence any) orthonormal basis $\{e_n\}_{n \in \mathbb{N}}$, the series $$s_k = \sum_{n=1}^\infty\big\...
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2answers
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$ tr(TS)=0$ and $T,S$ are positive, compact implies $TS=0$

Suppose $T$ and $S$ are positive compact operators and Tr$(TS)=0$ , then show $TS=ST=0$. Attempt: Apply the spectral decomposition for $T$ to obtain $\lambda_n(T)$ and corresponding orthonormal ...
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1answer
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How to prove that the trace is a linear functional? [closed]

Show that the trace function is a linear functional on the vector space of $n \times n$ matrices with entries in the field $\mathbb F$.
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What is the angle between a $A_{3x3}$ with $Rank(A)=2$ and $A^T$

I need to solve this problem: For Matrix $A_{3x3}$ with $Rank(A)=2$. If Matrix A is Transposed and its elements are the same as elements of Matrix B. What is the angle of rotation from A to B?
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1answer
48 views

Why does trace play such an effective role?

We know that trace plays a very important role in representation theory in the form of characters or in lie algebra in the form of killing form among many other things. What's so special about trace ...
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0answers
25 views

A formula for the trace of matrix monomials

I am looking for a general formula for the trace of monomials of the complex matrices $X=A+A^T$ and $P=i(-A+A^T)$, where $$ A=\begin{pmatrix} 0 &\sqrt{1} & 0 & 0 &...
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If A is an $n \times n$ matrix over the field $F$ with characteristic polynomial $f = (x - C_1)^{d_1} \dots (x - C_k)^{d_k}$

If A is an $n \times n$ matrix over the field $F$ with characteristic polynomial $$f = (x - c_1)^{d_1} \dots (x - c_k)^{d_k}.$$ What is the trace of $A$? My attempt: $A$ is similar to an matrix of ...
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1answer
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Prove that the inverse of trace of inverse is convex.

This is B.17 from Fundamentals of Convex Analysis by Jean-Baptiste Hiriart-Urruty, Claude Lemaréchal. Let $f: S^{++}(\mathbb{R}^n) \to \mathbb{R}$ be $$f(M) := \frac{-1}{tr(M^{-1})}$$ Then show $f$ ...
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Do the trace and norm always belong to the base field?

Let $F$ be a field and $E$ be a finite field extension of $F$. Let's call $F$ the base field. Do the two following proposition involving norm and trace hold? $\forall u \in F, N_{E/F}(u) \in F$. $\...
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1answer
39 views

Inequality for trace of product of matrices

Assume that $A \in \mathbb{R}^{n \times n}$ is a symmetric matrix and $B \in \mathbb{R}^{n \times n}$ is a symmetric positive definite matrix. Is the following statement true $$ \lambda_{\mathrm{min}} ...
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1answer
31 views

Existence of diagonal matrix that transforms invertible matrix in unitary

This questions arises from the answer to this previous question, which leaded me to 'relax' the statement that I wanted to prove. Suppose that a matrix $M\in\mathrm{SL}_2(\mathbb{C})\setminus\{\pm I\}...
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1answer
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Exercise about the trace of sobolev functions

Let $Q_1$ and $Q_2$ be two open squares in $\mathbb{R}^2$ whose closures have an edge - say L - in common. Let be $u_i\in W^{1,p}(Q_i)$ for $i=1,2$ and for such $p\in[1,+\infty]$. Suppose that $Tr(...
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1answer
33 views

Unitary matrix by conjugation

Suppose that a matrix $M\in\mathrm{SL}_2(\mathbb{C})\setminus\{\pm I\}$ has diagonal elements complex conjugates of each other and that their product is less than $1$. That is, $$M=\begin{pmatrix} a &...
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1answer
34 views

maximum of trace operator and 0

Let $ n \in \mathbb{N}, \ \Omega \subset \mathbb{R}^n $ be a bounded domain with Lipschitz-boundary and $ S:H^1(\Omega) \to L^2(\partial \Omega) $ the trace Operator. For $ u \in H^1(\Omega): \quad$ ...
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2answers
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How could a scalar be equal to the trace of the same scalar?

A scalar is supposed to be a matrix and a trace is suppose to be a number; how could the two be equal in the picture below where $Y$ is an $n \times 1$ random vector and $A$ is any $n \times n$ ...
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1answer
36 views

Traciality of compressions of von Neumann algebras

Let $\phi_1$ be a linear functional on a von Neumann algebra $\mathcal{A}.$ (I need the result in particular for $\Pi_1$-factors), satisfying traciality. With "traciality" I mean the following: For $...
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1answer
39 views

Trace of a the inverse of a complex matrix

I require the following $${\rm Tr}({\bf H} + \omega{\bf I})^{-1}$$ where $\bf H$ is Hermitian, $\omega$ is a complex number with a small imaginary component $(\Im \omega \approx 1\times 10^{-5})$ and $...
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0answers
32 views

minimise the trace of a matrix over all column permutations

I have a 10x10 positive symmetric matrix, I need to find the optimal permutation of the columns in order to minimise the trace. I can't try all permutations because that would be a 10! problem. Any ...
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1answer
66 views

$ \langle A, B\rangle := \operatorname{Tr}(AB^{∗}) $ Inner Product in $\mathbb{C} $?

We define $\langle\cdot, \cdot\rangle $: $\mathbb{C}^{n\times n} \times \mathbb{C}^{n\times n}\longrightarrow \mathbb{C} $ by $ \langle A, B\rangle := \operatorname{Tr}(AB^{∗})$. Show that this ...
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4answers
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How to choose $B$ to have $ \operatorname{trace}(H) \ge \operatorname{trace}(B^{\top}HB)$?

Considering $H \in \mathbb{R}^{N \times N}$ a real symmetric matrix which has both positive and negative eigenvalues, and $B \in \mathbb{R}^{N \times M}$ a real matrix with positive entries. Can I ...
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112 views

Show that the $\text{Tr}(A)^2 = \text{Tr}(A^2)+\text{Sum of Eigenvalues} $

Let $A$ be a square $ m \times m $ with eigenvalues $\lambda_{i},...,\lambda_{m}$. Show that: $$ [\text{Tr}(A)]^{2} =\text{Tr}(A^{2}) + \sum_{i \neq j} \lambda_{i}\lambda_{j} $$ Here is my attempt: ...
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1answer
73 views

How to show Von Neumann Trace inequality $ \text{Tr}(AB) \leq \sum_{i=1}^n \sigma_{A,i}\sigma_{B,i} $?

Let $A,B$ have the appropriate size. How can we show Von Neumann Trace inequality $ \text{Tr}(AB) \leq \sum_{i=1}^n \sigma_{A,i}\sigma_{B,i} $? Also, what is the intuition behind this inequality?
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2answers
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How to show $\text{Tr}(AB) \leq \|A\|_F\|B\|_F$?

Let $A,B$ be two $n \times n$ matrices. How to show $\text{Tr}(AB) \leq \|A\|_F\|B\|_F$. My try: Using Von Neumann trace inequality we have $$ \text{Tr}(AB) \leq \sum_{i=1}^n \sigma_{A,i}\sigma_{B,...
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2answers
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How to show $\sqrt{\text{Tr}(A^2)} \leq \text{Tr}(A)$?

Let $A$ be a positive semi-definite matrix. How to show that Frobenius norm is less than trace of the matrix? Formally, $$\sqrt{\text{Tr}(A^2)} \leq \text{Tr}(A)$$ Also, show when $A$ is an $n \times ...
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0answers
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Graded Trace Map

Let $k$ be a field and let $V$ be a finite dimensional $k$-vector space. Let $\Lambda_* : k \text{-vect} \rightarrow k \text{-alg}$ be the exterior algebra functor, whose target is associative $k$-...
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2answers
134 views

Let $A\in M_n(\mathbb{Q})$ with $A^k=I_n$. If $j$ is a positive integer with $\gcd(j,k)=1$, show that $ \operatorname{tr}(A)= \operatorname{tr}(A^j)$. [closed]

Let $A\in M_n(\mathbb{Q})$ with $A^k=I_n$. If $j$ is a positive integer with $\gcd(j,k)=1$, show that $ \operatorname{tr}(A)= \operatorname{tr}(A^j)$. I don't know how to start to prove that. ...
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1answer
24 views

Can an upperbound constraint on the squared Frobenius norm of a matrix be expressed as a linear matrix inequality?

Is it possible to express an inequality constraint on the squared Frobenius norm of a matrix $X$: $$\|X\|_F^2 = \mathop{tr}( X^T X ) \le t$$ as a linear matrix inequality? I want to say that it's: ...
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3answers
30 views

Dimension of $W$ Where $W$ is the subspace of matrices with trace=0

Let $V$ be the vector space of $n*n$ symmetric matrices.And let $W$ be the subspace of $V$ consisting of those Matrices $A$ such that $tr(A)=0$. What is the dimension of $W$? What I have noted is ...
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1answer
36 views

How to show for any symmetric matrices the quadratic mean of eigenvalues less than square of Frobenius norm?

Let $A$ be a symmetric matrix which has $k$ non-zero eigenvalue. Show that the square of Frobenius norm is always bigger than the average of squared eigenvalues. That is: $$\|A\|_F^2 \geq \frac{1}{k} ...
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1answer
62 views

What are the conditions on $\text{tr}(AB) \leq \text{tr(A)} \text{tr(B)}$ to be true?

Let $A$ and $B$ be two arbitrary matrix with proper dimension for multiplication. Consider this trace inequlaty which is trace of multiplication of two matrices versus their individual traces $$\...
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1answer
35 views

How to show $\|x-y\|_2^2 \leq \|x\|_2^2+2|x^Ty|$?

Let $x,y \in \mathbb{R}^n$. How can I show the following $$\|x-y\|_2^2 \leq \|x\|_2^2+2|x^Ty|$$ The above has been used by the authors of the following paper on page 8, in first line Online ...
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0answers
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Divergence theorem for inner product of Tensor

Let $\Omega\subset \mathbb{R}^3$ be bounded domain and $f$ be smooth vector field on $\Omega$ and $f|_{\partial\Omega=0}$. I not able to compute $\int _{\Omega} grad~f:(grad~f)^{T}$, where $:$ is ...
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1answer
27 views

UHF-algebra tracial state is faithful?

It is known that a unital UHF-algebra has a unique tracial state, it is true that it is true that this trace is normal and faithful? I am particularly interested in the universal UHF-algebra, i.e. the ...
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2answers
38 views

Trace of product of gradient

Consider vector field $v$. We know, that $Tr(\nabla v)=div ~ v$. Is is true that $$Tr(\nabla v\nabla v)=(div ~ v)^2$$ Thanks for any hint.
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0answers
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Solving matrix equation containing trace and two-sided products

Let $A,B,Y$ be matrices and $d$ be a scalar. Is there an analytic solution for $A$ in the following expression? $$(A+B)^{-1} Y (A+B)^{-1} - (A+B)^{-1} + \frac{d A^{-2}}{\operatorname{tr}( A^{-1} + B^{...
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derivative of an integral with respect to matrix element

I need to compute the following derivative: $$ \frac{\partial}{\partial V_{mn}} \int \prod_j d\phi_j \, \mathrm{Tr} \left(\ln(i \phi + \beta J V)\right) $$ where $\phi \equiv \phi_i \delta_{ij}$ is a ...
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45 views

Trace of a product of two positive definite matrices

Let $A, B, C_t \in \mathbb{R}^{n \times n}$ be positive definite matrixand $C_t$ is defined as \begin{align*} C_{t,(i,j)}=\begin{cases} A_{i,j}, &i,j < t \\ B_{i,j}, &i,j \ge t \\ 0, &...
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1answer
67 views

Prove that $tr(A)^p = tr(A^p)$ $mod$ $p$ where $A$ is a square integer matrix and $p$ is a prime number.

I'm looking for an elementary proof (one which does not use Galois theory). For the case $p = 3$, we have that $tr(A^3) = tr(A)^3 - 3e_1e_2 + 3e_3$ where the $e_i$ are coefficients of the ...
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1answer
86 views

If $A$ is an $n$ by $n$ integer matrix such that $A^3 = I$, then $\operatorname{tr}(A) = n\mod3$

Attempt: We work with $A'$, the matrix with entries $a_{ij}\mod 3$. Note that cubing $A'$ still gives $I$ as $I$ is unchanged by considering remainders $mod$ $3$. For the rest of the proof, we will ...
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1answer
51 views

Given this 4x4 matrix, is there an easier way to calculate the eigenvalues?

This came up as a textbook question: Find the rank and 4 eigenvalues of A, where A is the 4x4 matrix with all 1 entries 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 I am ...
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0answers
37 views

Proving an inner product

Let $V$ be the vector space of all $2$ by $2$ matrices. Define $\langle A, B\rangle = \mathrm{tr}(A^TDB)$, where $D = \begin{pmatrix}7&1\\ 1&1\end{pmatrix}$. Prove that $<A, B>$ defines ...
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1answer
63 views

Prove that $r(A)=\operatorname{tr}(A^2)$

Let $A\in M_n(\mathbb{C})$. Show that if $A^3=A$, then $r(A)=\operatorname{tr}(A^2)$. Since $A^3=A$, the possible eigenvalues are $0,1,-1$. I don't know from here how to compute the rank of $A$. ...
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1answer
14 views

Equality of tracial states on dense $C^*$-subalgebra implies equality on generated von Neumann algebra?

Maybe this is a simple question, but I'm not sure about the following: Let $\cal M$, $\cal N$ be von Neumann algebras and $X\subseteq \cal M$ a weakly dense (possibly separable) $C^*$-subalgebra. Let ...