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Questions tagged [trace]

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

2
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1answer
30 views

If $A \in C^{nxn}$ , $A \ge 0 $ and A is sing., there exists a sequence of matrices $C_k$, that $C_k \ge 0$,$|C_k| = 1$ and trace $AC_k \le 1/k$

Question: Show that if $A \in C^{n \times n}$ , $ A \ge 0 $ and A is singular, then there exists a sequence of matrices $C_k$, $k = 1,2,...$ such that $C_k \ge 0$, det $C_k = 1$ and trace $AC_k \le 1/...
2
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1answer
40 views

How to show $\text{rank}(p)=\text{trace}(p)$ for every projector $p$ defined on $\mathbb{R}^n$? [duplicate]

I know this is an old question and there are several answers for this using eigenvalues and matrix factorization but they have not taught in my matrix analysis course yet. Therefore, my question would ...
0
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1answer
27 views

Is the trace of the matrix obtained by subtracting two positive definite matrices smaller than the trace of the matrix being subtracted?

Let $A$, $B$ be symmetric positive definite matrices. If $C=A-B$, is it true that $$ \operatorname{trace}\left(C^{-1}\right) > \operatorname{trace}\left(A^{-1}\right)\:?$$
5
votes
1answer
60 views

Rewrite $(\det A)^{1/n}=\min\left\{\frac{\mathrm{tr}(AC)}{n}:C \in \Bbb{C}^{n×n},C>0,\det C=1\right\}$ in terms of $\frac{\rm{tr}(CAC)}{n}$

Given $$ (\det A)^{1/n} = \min \left\{\frac{\operatorname{tr}(AC)}{n} : C \in {\Bbb C}^{n \times n}, C > 0, \det C = 1\right\}. \label1\tag1 $$ Question Show that the formula can be rewritten as $...
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0answers
30 views

compute $H^{3/2}(\partial\Omega)$-norm for smooth $u$ and $\Omega$

I am a little bit confused about different definitions of the trace space $H^{3/2}(\partial \Omega)$, and I hope I can find some simple examples on how to explicitly compute these norms for simple ...
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0answers
20 views

relation between trace and hat operator (skew-symmetric matrices)

To avoid confusion, let me first introduce the notation (although pretty standard) which is required for the question that I want to ask. Let $\mathsf{GL}(3,\mathbb{R})$ be the set of $3\times 3$ real ...
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0answers
11 views

Finding the partial of a quadratic weighted $\mathbf{Q}(\mathbf{x})$?

If $\mathbf{Q}(\mathbf{x})$ such that $\mathbf{x}\in\mathbb{R}^m$ and $\mathbf{Q}\in\mathbb{R}^{n\times n}$ is the following logic correct? \begin{align*} \frac{\partial\mathbf{u}^T\mathbf{Q}(\mathbf{...
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votes
1answer
49 views

If sum of traces of matrices at k-th power is 0, eigenvalues=0?

Given $$A_1^k + A_2^k + \cdots + A_m^k = 0, \qquad \forall k \in \mathbb N^+$$ then $$\mbox{Tr}(A_1^k) + \mbox{Tr}(A_2^k) + \cdots + \mbox{Tr}(A_m^k) = 0$$ where $A_1, A_2, \dots, A_m$ are $n\...
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0answers
17 views

$U_{p,q}$ is bounded

I am trying to prove $U_{p,q}$ is bounded using the induced norm $|| . ||_2$ from $M_n(\Bbb R)$ (or $M_n(\Bbb C)$ I am not sure). A norm is an application $M_n(\Bbb C) \to \Bbb R^+$, but in the case ...
1
vote
1answer
23 views

Trace and Det of Laplacian on the rectangle

I consider the eigenvalue problem $\Delta \varphi = \lambda \varphi$ with the Dirichlet boundary condition $\varphi|_{ \partial \Omega}=0$ on the rectangle $\Omega= [0,l] \times [0,m]$. By using ...
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0answers
18 views

Expressing a subtraction from a Hermitian outer products' main diagonal in terms of the original vectors

Consider a vector $\mathbf{x} \in \mathbb{C}$, and its complex conjugate transpose $\mathbf{x}^H$. Computing the outer product of both vectors results the matrix $\mathbf{\Phi} = \mathbf{x} \mathbf{x}^...
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1answer
69 views

Product rule for trace of matrix functions

I am trying to find the gradient of $f(Z_2) = \|A - Zg(Z_1g(Z_2)) \|_F^2$ with respect to $Z_2$ where $g$ function is applied to each matrix element wise such that $i,j$ element of matrix $g(X) = g(X_{...
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2answers
16 views

Confused about matrix tracing

I need help with this one question I found from my Linear Algebra textbook. While I was doing problems and checking my work from the back of my textbook, I found out one of the problems I did is wrong....
5
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1answer
75 views

What is the Largest Dimension of the Set of Matrices with $\text{trace} AB = 0$

Suppose $W$ is a subspace of $M_n(\mathbb{R})$ with property that $\text{trace}(AB) = 0$ for all $A,B \in W$. I want to find the largest possible dimension of $W$. It seems like the answer is $n(n - ...
3
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3answers
86 views

Prove $tr(A^2) \leq tr(A^TA)$

Qu: prove $tr(A^2) \leq tr(A^TA)$ I saw the below link before and think it is related to this question $A,B$ be Hermitian.Is this true that $tr[(AB)^2]\le tr(A^2B^2)$? but still can't solve it. I ...
0
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1answer
92 views

An inequality on trace of sum of matrices

Let $A$ be a stable matrix (that is, $\rho(A) <1$) and $B$ be a symmetric positive semi-definite matrix. Now, if we define $$ f(N) = \sum_{i=1}^N \mbox{tr} \big((A^{i-1})' B A^{i-1} \big), $$ can ...
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0answers
22 views

Trace of product of squared matrix and positive definite matrix is nonnegative (short exercise)

Let $\Sigma$ be a real $d \times d$-matrix and $C$ be a real, symmetric, positive definite $d \times d$-matrix. Does it then hold, that $$ \text{tr}(\Sigma^2C) \geq 0 \quad ?$$
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1answer
26 views

Question on trace Sobolev's theorem for domain $\Omega \times (0,T)$

Let $\Omega \subset \mathbb R^3$ be an open,bounded subset with a $C^2-$boundary $\Gamma$. Fix $T>0$. Can we claim that $W^{1,2}(\Omega \times (0,T)) \hookrightarrow C([0,T];L^2(\Gamma))(*)$ ...
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1answer
36 views

How to simplify this trace term?

I have the following trace term: trace(Sk' Ck Sk) where Sk is a KxM matrix and Ck is a KxK positive semidefinite matrix. I'm involving this trace term in a ...
0
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1answer
44 views

Prove that $A$ and $B$ are proportional

Let $A$ and $B$ are $n \times n$ matrices. Also, $$|\operatorname{tr}(A^TB) \,|^2 = \operatorname{tr}(A^TA) \, \operatorname{tr}(B^TB)$$ Prove that $A$ and $B$ are proportional. I don't know with ...
0
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1answer
46 views

How is the trace of the adjoint of Lie algebra elements defined? [duplicate]

Consider an element $X\in\mathfrak g$ of some Lie algebra $\mathfrak g$. I understand that $\mathfrak g$ can be represented via its action on other elements of the same algebra, as $\operatorname{ad}(...
0
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1answer
41 views

Proof that eigenvalues of matrix with zero trace are all equal to zero

I'm working on the following question: "Suppose that $A$ is a complex square matrix such that the trace of $A^k$ is zero for every $k \in \mathbb{N}$. Show that all the eigenvalues of $A$ are zero." ...
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0answers
38 views

Matrix product and eigen values

Is there any relationship between eigenvalues(or spectrum) of graph Laplacian matrix and the eigenvalues of the product of a real symmetric matrix and the Laplacian matrix? My problem at hand is as ...
0
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1answer
68 views

Differentials to derivatives involving trace of matrices

Suppose $P$ is a real-valued function of the $p\times m$ (real) matrix $\mathbf{Q}$. After taking its differential, one arrives with the following: $$ d(P(\mathbf{Q})) = \operatorname{trace}\...
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1answer
82 views

Partial Derivative of Trace of Matrix in negative power wrt to parameters

$\renewcommand{\v}[1]{\mathrm{vec}\left(#1\right)} \renewcommand{\m}[1]{\mathbf{#1}} \renewcommand{\trace}[1]{\mathrm{trace}\left(#1\right)} \renewcommand{\diag}[1]{\mathrm{diag}\left(#1\right)}$ ...
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0answers
12 views

How to compute $cov(\frac{1}{2}Z'AZ,\frac{1}{2}Z'BZ)$?

I want to show that $cov(\frac{1}{2}\textbf{Z}'A\textbf{Z},\frac{1}{2}\textbf{Z}'B\textbf{Z})=\frac{1}{2}tr(AR(\theta)BR(\theta))$ where $A,B \in Sym(p)$ (real symmetric $p \times p$ matrices) and $\...
2
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1answer
32 views

Free probability and projections

Let $(\mathcal{A},\varphi)$ be a free probability space, where $\mathcal{A}$ is a von Neumann algebra and $\varphi$ a finite and faithful trace. Let furthermore $p\in\mathcal{A}$ be a projection. ...
1
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1answer
27 views

Convergence of finite dimensional projection of trace class in trace norm

Assume $\mathbb{H}$ is a Hilbert space and $K$ is a trace-class operator on it. Given a fixed ONB $\{e_i\}$ and assume $$K=\sum_{i,j}c_{ij}e_i\otimes e_j.$$ Now, let $K_n = \sum_{1\leq i,j\leq n}c_{...
0
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1answer
39 views

Given two square matrices $A_{nxn}$ and $B_{nxn}$, prove that $trace (\mathbf{A}^\intercal \mathbf{B}) = trace (\mathbf{A} \mathbf{B}^\intercal)$.

I checked with an example of random matrices and I noticed the sum of the resulting diagonals are indeed the same. Also, I have that $trace (\mathbf{A}^\intercal \mathbf{B}) = trace(\mathbf{B} \...
1
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1answer
84 views

Matrix derivative of $Tr(A\log(X))$

I'm trying to work out the derivative of $Tr(A\log(X))$ with respect to $X$. Assume $X$ is positive so the $\log$ is well defined. I know that $$Tr(A\log(X)) = A^\dagger: \log(X)$$ but what I ...
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0answers
27 views

Triangle Inequality of Tensor Products

If $$\|A - x\|_1 \le \epsilon$$ and $$\|B - y\|_1 \le \epsilon$$ where $A, B, x, y \in Herm(H_A)$, where $Herm(H_A)$ are the set of Hermitian matrices in a Hilbert space $H_A$, then can we say, by ...
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0answers
29 views

Are the invariants related in the characteristic equation of an orthogonal matrix (3x3 matrix)?

The characteristic equation of matrix A is $$\lambda ^3 - I_1\lambda^2 + I_2\lambda-I_3 = 0 $$ For orthogonal matrix $$I_3 = det(A) = \pm1$$ $$I_1 = tr(A)$$ Taking examples of orthogonal matrices, ...
1
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2answers
26 views

Show: $\sum_{i,j=1}^n |a_{ij}|^2=trA^*A=\sum^n_{i=1}\sigma_i^2$

I'm trying to prove the above fact for an arbitrary matrix $A$, with eigenvalue $\lambda_i$, and singular values $\sigma_i$. My approach so far: the trace of a matrix is the sum of its eigenvalues, ...
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0answers
25 views

Show that any zero trace matrix is ​similar to a null diagonal matrix.

Show that any zero trace matrix is ​​similar to a null diagonal matrix. let $A=(a_{i,j})_{1 \leq i,j \leq n} \in M_n(K)$ such that $ \sum_{k=1}^{n} a_{k,k}=0$ I need to show that that there is $P \...
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0answers
21 views

Rewriting this matrix expression involving the trace operator

Let's assume the minimum necessary conditions for the following expression to be well-defined: $$-\frac{1}{2} \mbox{Tr} \left(\Sigma^{-1}\left(\Lambda+\Phi -\Psi W- W\Psi^\intercal+W(\Xi+V^{-1})W^\...
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1answer
27 views

Did author omit trace in this problem statement?

I have this problem: Prove that the following identity is true: $$\boldsymbol{A} \boldsymbol{A^T} = \sum_{i} \boldsymbol{a_i} \boldsymbol{a_i^T}$$ I assumed the notation $a_i$ means ith column ...
1
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2answers
29 views

How to approach this question on subspaces?

Determine which of the following are subspaces of $3 \times 3$ matrix $M$ all $3 \times 3$ matrices $A$ such that the trace of $A$ is $\mbox{tr}(A) = 0$. What does trace mean?
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1answer
44 views

Monotonicty of the trace on Sobolev spaces.

Let $\Omega \subseteq \mathbb{R}^n$ be a bounded domain with smooth boundary and suppose $u\in W_0^{1,p}(\Omega)$. If $v\in W^{1,p}(\Omega)$ satisfies $\lvert v \rvert \leq \lvert u \rvert$ in all of $...
1
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1answer
60 views

Trace of product of matrices with nonzero trace

I have the four matrices $$\begin{pmatrix}1&0&0&0\\1&0&0&0\\0&1&1&1\\0&1&1&1\end{pmatrix},\quad \begin{pmatrix}0&1&0&0\\0&1&0&0\...
3
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1answer
53 views

Maximal subgroup of permutations for which trace is invariant

Recall that trace is invariant under cyclic permutations of a product of matrices. Are cyclic permutations the maximal subgroup of permutations for which trace doesn't change?
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0answers
35 views

If and only if condition on $B$ when $\mathrm{tr}(AB)-\mathrm{tr}(A)\geq0$.

For $a_i,b_i\in\mathbb{R}^+$ for all $i=1,\cdots,n$, I want to find if and only if condition on $b_i$'s when \begin{equation} \sum_{i=1}^na_ib_i\geq\sum_{i=1}^na_i. \end{equation} To transform this ...
1
vote
1answer
40 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 $ tr(A)= tr(A^j)$.

I saw this question and answer at following site. enter link description here The answer at this site is here. Note that eigenvalues $\lambda_i, i=1,\ldots, n$ of $A$ are roots of the rational ...
0
votes
1answer
32 views

Finding a field extension in which every element has zero trace

Let F be a field extension of K, then F over K is a vector space, and for each a in F define f:F-->F as f(x)=ax, this is a linear transformation, define trace of a as trace of this linear ...
0
votes
3answers
41 views

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 $n \times n$-matrix $A$ with real matrix elements we have $A^2 = -E$. Prove that $\operatorname{tr}(A) = 0$. Any ideas ?
3
votes
2answers
85 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 ...
0
votes
1answer
131 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 ...
0
votes
1answer
29 views

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(...
2
votes
0answers
50 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)...
0
votes
1answer
19 views

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 ...
0
votes
1answer
39 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 ...