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

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0
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1answer
19 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
29 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
votes
0answers
9 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. ...
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0answers
35 views

how to compare trace($B^4$) and trace[$(B-D)^4$] with $D$ the diagonals? [closed]

If $B=(b_{ij})_{n\times n}$ is a real symmetric $n$ by $n$ matrix, $D = (d_{ij})_{n\times n}$, defined as $d_{ij}=b_{ij}$ if $i=j$ and $0$ otherwise. then how to compare $\text{Trace}(B^4)$ and ...
3
votes
1answer
66 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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3answers
33 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 ...
5
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1answer
50 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 $\lambda\in F$. ...
0
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0answers
26 views

Derivative of trace of matrix product

Provided $\mathbf{P}$ is invertible, Solve \begin{equation} \frac{\partial}{\partial\mathbf{D}} ...
4
votes
2answers
187 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
58 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
32 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
50 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
63 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
46 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
34 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\\ ...
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0answers
21 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 ...
3
votes
1answer
213 views

Evaluate $\int{\mathrm{tr}}\left( {AB(I-xB)^{-1}}\right) {\,dx}$ for $A$ and $B$ square real matrices

Let $A$ and $B$ be two real $n\times n$ matrices, and let $C(x)=B(I-xB)^{-1}$, where $I$ is the identity matrix of order $n$, for any real scalar $x$ such that $I-xB$ is invertible. Denote by ...
0
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0answers
19 views

matrix $A$ satisfy in $\operatorname{tr}(A^m)=0$, prove it is nilpotent matrix. [duplicate]

Let $A\in M_n(\mathbb{R})$ and for any $m\in \mathbb{N}$, $\operatorname{tr}(A^m)=0$. Prove $A$ is a nilpotent matrix.
4
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2answers
53 views

Polynomial of $3\times 3$ matrix with $\operatorname{tr}(A)=\operatorname{tr}(A^2)=0$ and $\det(A)=1$

If $A$ be an invertible matrix $M_3(\mathbb{R})$ and we have $\operatorname{tr}(A)=\operatorname{tr}(A^2)=0$ and $\det(A)=1$, then what is the characteristic polynomial of $A$?
0
votes
1answer
24 views

Linear Algebra proof involving traces and rank one projections

How does one prove that if $T$ is a non-negative linear operator over a finitely generated Hilbert space that satisfies $tr(T^2)=tr(T)=1$, then $T$ must be a rank one projection? I am sorry to give no ...
3
votes
1answer
25 views

What can we say about the trace of this matrix?

Consider $$ E = V\Lambda V^{\top} $$ $$ (m\times m) = (m\times p) (p\times p) (p\times m)$$ where $p<m$, $E$ is symmetric and singular, $\Lambda$ is diagonal, $V$ has orthonormal columns, and $$ ...
4
votes
3answers
68 views

How do I prove that the trace of a matrix to its $k$th power is equal to the sum of its eigenvalues raised to the $k$th power?

Let $A$ be an $n \times n$ matrix with eigenvalues $\lambda_{1},...\lambda_{n}$. How do I prove that tr$(A^k) = \sum_{i=1}^{n}\lambda_{i}^{k}$?
2
votes
1answer
34 views

Inequality on the trace of the resolvent of a matrix

For a (random) hermitian matrix $M$ and a complex $z$, it is well known that $$ \left| \int_{\mathbb{R}} \frac{1}{z-x} \text{d}\mu_M(x) \right| = \left| \frac{1}{n} \text{Tr} (z-M)^{-1} \right| \leq ...
1
vote
0answers
43 views

Weak convergence = norm convergence for trace class operators?

Given a (separable) Hilbertspace $H$, I look at the traceclass operators $\mathfrak{S}_1$. I recall the fact that the weak convergence implies norm convergence in the sequence space $\mathcal{l}^1$. ...
1
vote
2answers
60 views

Unitary matrix in trace and log function

I am trying to do a unitary transformation $U$ on square matrix $A$ which is embedded inside a trace and natural log function, and the following property is supposed to hold: $\mathrm{tr} (\ln (A)) = ...
2
votes
1answer
22 views

Intuition of the cyclic trace property of matrix products and relation to the determinant?

Let $A_1,\ldots,A_n$ be square matrices (not necessarily symmetric). Taking the trace of a matrix product and a cyclic permutation of the product yields: \begin{align*} \text{trace}(A_nA_{n-1}\cdots ...
1
vote
1answer
20 views

Partial derivatives of the marginal likelihood of a Gaussian Process

In Chapter 5 of "Gaussian Processes for Machine Learning" by Rasmussen and Williams on page 114 (p.10 in pdf) they give the equation (5.9) to calculate the partial derivatives of the marginal ...
0
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3answers
34 views

Traces and Eigenvalues $3\times 3 $ matrix

I understand how to find the trace of a $2\times 2 $ matrix but I am confused with how to find the characteristic polynomial of a general $3\times 3 $ matrix with arbitrary constants.
1
vote
1answer
22 views

adjoint operator of the partial trace map

Could someone explain to me, what is the adjoint map of the partial trace map the (tensored with the identity map), or why does the following equality hold? $Tr(C_A\cdot Tr_{B} D_{AB})=Tr((C_A\otimes ...
3
votes
1answer
56 views

Finding a symmetric 3x3 matrix from 2 eigenvectors and 2 eigenvalues

Good evening everybody, I'm stuck with the following problem from an old exam in Linear Algebra. One is given two Eigenvectors with corresponding two Eigenvalues and told that the trace is negative. ...
1
vote
1answer
49 views

Derivative of the trace of the power of a PD matrix

Suppose $\mathbf{X}$ is a $n\times n$ positive definite matrix, $\mathbf{A}$ is a $n\times n$ constant matrix, and $b$ is a real scalar. The matrix power $\mathbf{X}^b$ is defined to be ...
0
votes
1answer
19 views

The norm of trace of functions in $H^\frac{1}{2}(\partial\Omega)$

Let $\textbf{A}\in(H^1(\Omega))^3$, where $\Omega\subset\mathbb{R}^3$ is a bounded convex domain with its boundary $\partial\Omega$. Now we know, on $\partial\Omega$, ...
0
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0answers
15 views

Is this enough to ensure $\mathrm{tr}(z^\dagger z)<2$?

Let $z $ be $n\times n$ skew-symmetric complex matrices, satisfying the Plücker relations $$ z_{ij}z_{kl}=z_{ik}z_{jl}+z_{il}z_{jk} $$ and such that $\mathbb{I}_n-z^\dagger z>0$, i.e. the ...
7
votes
2answers
112 views

Problem involving trace and determinant of symmetric matrices

I've stumbled upon this exercise on a linear algebra book that asks me to determine all the ordered pairs $(a,b)$ of real numbers to which there exists an unique symmetric matrix $A\in R^{2\times 2}$ ...
0
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0answers
22 views

solution to recurrence relation involving terms of the form $\mathrm{tr}[(z^\dagger z)^k]$

While working on my research I encountered a recurrence relation which I am having trouble solving. The problem is the following: let $z$ be a $n\times n$ skew-symmetric (but not necessarily ...
2
votes
1answer
28 views

Convex Optimization - nuclear norm regularisation of symmetric matrix

I have a problem of the form $\min_{X\in \mathbb{R}^{n \times n}} g(X) - \lambda ||X||_{*}$ where $g$ is convex and differentiable. I would like to use proximal gradient descent to solve this. How ...
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0answers
41 views

How to prove $\partial [\mathbf{Z}^{\mathrm{T}} \boldsymbol{x}] / \partial \mathbf{Z} = \boldsymbol{x}$

I have an $M \times N$ matrix $\mathbf{Z}$ and a $M \times 1$ vector $\boldsymbol{x} = \boldsymbol{n} / N$, where $N = \boldsymbol{1}^{\mathrm{T}} \boldsymbol{n}$. I want to calculate the total ...
0
votes
1answer
34 views

Find an integer $c$ that satisfies the equation $|A|={1\over c}[(A)^3-3tr(A)+2tr(A^3)]$ where $A\in M_{3x3}(\mathbb{R})$.

Find an integer $c$ that satisfies the equation $|A|={1\over c}[tr(A)^3-3tr(A)tr(A^2)+2tr(A^3)]$ where $A\in M_{3x3}(\mathbb{R})$. I know that $c=6$ from wikipedia, but I don't know how to show it. ...
2
votes
1answer
37 views

Correct way to calculate a matrix trace with negative values

I have a 10x10 symmetrical variance-covariance matrix, such that the variances for 10 vectors are on the main diagonal and the covariance between all vectors are on the off-diagonals. I want to ...
0
votes
1answer
90 views

Let $A,B ∈ M_{n×n}(F)$ be matrices satisfying $A^2 + B^2 = I$ and $AB + BA = O$. Show that $tr(A) = tr(B) = 0$.

Let $F$ be a field and let $n$ be a positive integer. Let $A,B ∈ M_{n×n}(F)$ be matrices satisfying $A^2 + B^2 = I$ and $AB + BA = O$. Show that $tr(A) = tr(B) = 0$. I know that $tr(A^2)+tr(B^2)=n$ ...
0
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0answers
14 views

Automorphisms of the exceptional Jordan algebra preserve the determinant and trace.

I am trying to show that the automorphism of the exceptional Jordan algebra $\mathbb{J}_3(\mathbb{O})$ preserve the determinant and trace. This algebra consists of $3x3$ Hermitian matrices over ...
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0answers
46 views

Sum of traces over Weyl group

I'm interested in computing sums like $\sum_{\sigma \in W} tr(\sigma ^3)$ , where $W$ is the Weyl group of $SO(2n+1)$, i.e. $W = (\mathbb{Z}/2\mathbb{Z})^n \rtimes S_n$. I tried to figure out what an ...
0
votes
1answer
23 views

Existence of matrix $C$ such that $f(A)=tr(AC)$

Let $f: M_{n\times n}(\mathbb R)\to \mathbb R$ a functional. Prove that there exists a unique square matrix $C$ such that $f(A)=tr(AC)$ for all $A\in M_{n\times n}(\mathbb R)$ I´ve been trying to ...
1
vote
1answer
48 views

To compute the trace of a linear transformation

Let $A$ be the matrix, $$A=\begin{bmatrix} 3 & 0 & 0 & 2\\ 0 & 3 & 2 & 0\\ 0 & 2 & 3 & 0\\ 2 & 0 & 0 & 3 \end{bmatrix}$$ And the linear transformation ...
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vote
0answers
18 views

Interchange of trace and infimum

Let be such $f( \cdot,t): \mathbb{R}^{n \times n} \to \mathbb{R}^{n \times n}$, where $t \in \mathbb{R}$. How to show that the following is true \begin{align} \text{Trace } \left(\inf_t f(X,t) ...
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0answers
29 views

Matrix Multipication and Inner Product

I am wondering about the difference between matrix multiplication and inner product. It is regarding the following question: Let $N \in M^{\Bbb C}_{n \times n}$ be a normal matrix. Prove that if ...
1
vote
1answer
18 views

Trace of the product of a rank-one and an indefinite matrix, subject to semidefinite constraints

Let $Q$ be a Hermitian (indefinite) matrix. Is it true that \begin{equation} \operatorname{tr}(QX)\geq0 \quad\text{and}\quad X-xx^T\succeq0 \quad\Rightarrow\quad \operatorname{tr}(QX) ...
0
votes
1answer
24 views

Representative Matrix for a Bilinear Form where $M=\left( \begin {matrix} 2 & 4 \\ 8 & 6\end {matrix}\right)$

I don't understand how to solve the following question, and maybe the basic idea of it. Let $V=M^R_{n \times n}$ and let $f: V \times V \to \Bbb R$ define as $f(A,B)=tr(A^tMB)$, for each $A,B \in ...
2
votes
2answers
131 views

trace of symmetric matrix problems

I have the two problems below from a practice exam. I can prove them on their own but am not exactly sure if/how to show that they only hold for symmetric matrices and for '3)' showing that it only ...
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0answers
14 views

Why do we divide out by commutators for the trace operation?

Given a unital ring $R$, it's easy to form the ring $M_n(R)$ of square $n\times n$ matrices over $R$. Any such matrix has a trace, which is simply the sum along the diagonal. However, I've come to ...