Questions tagged [trace]
For questions about trace, which can concern matrices, operators or functions. If your question concerns the trace map that maps a Sobolev function to its boundary values, please use [trace-map] instead.
1,743
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Trace of matrix product involving identity and powers
Suppose we know $\text{Tr}(A)=a$. Is there a closed formula for obtaining
\begin{eqnarray}
\text{Tr}(A(A-I)^n),
\end{eqnarray}
for any $n=1,2,...$ and with $I$ being the identity matrix? Such products ...
- 448
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1
answer
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Is it true that $tr(APA^T)$ > $tr(AQA^T)$, if $tr(P)$ > $tr(Q)$
Assuming $P$ and $Q$ and positive definite matrix.
Is it true that $tr(APA^T)$ > $tr(AQA^T)$, when $tr(P)$ > $tr(Q)$. [EDIT after first answer: not just that, actually $P_{ii} > Q_{ii} $ for ...
- 149
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4
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To find the trace and determinant of a matrix $A$ satisfying $A^{2023} + A = \left(\begin{smallmatrix}2 &2& 0\\ 0&2&2\\ 0&0&2\end{smallmatrix}\right)$ [closed]
If $A$ is a $3 \times 3$ matrix such that
$$
A^{2023} + A = B\quad \mbox{where matrix}\ B\
\mbox{is given by}\quad B =
\left(\begin{smallmatrix}2 &2& 0\\ 0&2&2\\ 0&0&2\end{...
- 21
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24
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Minimize $\mathrm{tr}((FK + G)^TQ(FK + G) + K^TRK)$ over block lower-triangular matrices $K$
I want to solve the minimization problem
$$
\inf_{K\in\mathcal{K}}\mathrm{tr}\left(\left((FK + G)^TQ(FK + G) + K^TRK\right)\Sigma\right)
$$
where $\mathcal{K}$ is the set of block lower triangular ...
0
votes
1
answer
54
views
Norm of a simple extension
We have the following setup: $K$ a field $L= K( a)$ algebraic field extension and $m_a$ its minimal polynomial. We need to show that for each $x\in K$ we have: $m_a(x) = N_{L/K}(x-a)$.
I just plugged ...
- 423
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Divergent Tail Sums of Approximations of Non-trace Class Compact Operators
I'm working on approximations of compact operators that are not trace class, and I'm looking for ways to provide meaningful approximation error estimates for truncated eigenfunction expansions. I ...
user1328882
3
votes
2
answers
91
views
$f=0$ on $\partial\Omega$ implies $f\in H_0^1(\Omega)$
Let $\Omega\subset\mathbb{R}^n$ be an open set. Let $f\in C(\bar{\Omega})\cap H^1(\Omega)$ with $f=0$ on $\partial\Omega$.
Claim: Then $f\in H_0^1(\Omega)$ holds.
Since $H_0^1(\Omega)$ is the closure ...
- 209
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votes
1
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views
Finite dimensional Irreps (of algebras) with same traces must be equivalent ('page 136' in Bourbaki)
I look for the reference (or proof) of the following fact which is from appendix (B $27$) of Dixmier's book on $C^*$-algebras.
Claim: Let $A$ be an algebra (not necessarily commutative) over a field $...
- 1,589
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Double Trace of the tensor product of the metric tensor with vector fields.
So I am currently preparing for an exam on General Relativity and while reading the notes I stumbled upon this:
$$
tr[tr[g \otimes X \otimes Y]]= g(X,Y)
$$
Where
$$
g=g_{ij} dx^{i}\otimes dx^{j}
$$
is ...
- 3
1
vote
0
answers
52
views
Trace morphism in deligne/milne:s "tannakian categories".
Is there a connection between the trace morphism (1.7.3 on page 10 in https://www.jmilne.org/math/xnotes/tc2022.pdf) and characters of finite groups?
I am also trying to understand why 1.7.4 on the ...
- 1,276
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50
views
An "almost" true inequality for Hermitian matrices
Let $A$ be an $N\times N$ Hermitian matrix. For $p+q$ even, consider the following inequality:
$$\frac{1}{N}\sum_{i=1}^N (A^p)_{ii} (A^q)_{ii} \geq \Big(\frac{1}{N}\sum_{i=1}^N (A^p)_{ii} \Big) \Big(\...
- 422
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Proof of Conjecture on `Block-Orthogonalisation Always Reduces Trace'
I have a symmetric positive definite matrix, ${\bf D}\in\mathbb{R}^{NK\times NK}$, which is made up of $K$, $N\times N$, blocks:
\begin{equation}
{\bf D} = \begin{bmatrix}{\bf D_{11}} & {\bf D_{12}...
- 167
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Trace of Operators [closed]
Let $H_1$ and $H_2$ be two Hilbert Spaces. Let A be an bounded linear operator between $H_1$ and $H_2$ such that $AA^*$ is traceclass, where $A^*$ denotes the adjoint Operator.
Is it true that we have ...
- 11
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3
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Inequality involving matrix trace and diagonalisable matrices
Given two real PSD matrices diagonalisable by orthogonal matrices: $A=UDU^T$ and $B=VEV^T$, prove that
$$tr(A+B-2(A^{1/2}BA^{1/2})^{1/2})\geq0.$$
We can rewrite the inequality as
$$tr(D)+tr(E)\geq tr((...
- 991
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Finding general expression for symmetric trace-free tensors (STFs)
Given a symmetric $n$-tensor $I_{\alpha_1 ... \alpha_n}$, its tracefree version is given by
$$Q_{\alpha_1 ... \alpha_n}=\sum_{k=0}^{\left\lfloor{\frac{n}{2}}\right\rfloor} (-1)^k \frac{ \binom{n}{k} \...
- 265