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.
1,779 questions
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Prove that $tr(A^4)=8q-2m+2\sum_{i=1}^{n=\vert V(G)\vert}\deg^2(v_i).$
Let $G$ be a graph with adjacency matrix $A$. Prove that $$tr(A^4)=8q-2m+2\sum_{i=1}^{n=\vert V(G)\vert}\deg^2(v_i),$$ where $q$ is the number of $4$-cycles in $G$ and $m$ is the number of edges in $G$...
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canonical involution in group algebras
Let $G$ be a finite group. If $K$ has involution $-:K \to K$, the canonical involution $*:KG \to KG$ is defined as $$(\sum_ga_gg)^*=\sum_g\bar{a}_gg^{-1}.$$
Question: When is each centrally primitive ...
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Canonical trace on simple algebras
Let $A$ be a finite-dimensional simple $K$-algebra. The map $$T:A \to K; \ \ T(a)=tr(l_a),$$ where $\text{End}_K(A) \ni l_a:x \mapsto ax$ and $tr$ is the ordinary trace of linear operators, is called ...
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Find a formula for $\operatorname{tr} T.$
Here is the question I am thinking about:
Suppose $V$ is an inner product space and $v,w \in V.$ Define an operator $T \in \mathcal{L} (V)$ by $Tu = \langle u,v \rangle w.$ Find a formula for $\...
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Fidelity inequality of Covariance matrix and its block diagonal version
Let $F(A,B)=\text{Trace}((A^{1/2}BA^{1/2})^{1/2})$ be the fidelity between two PSD matrix $A$ and $B$. The fidelity is a quantity describing how close the two matrices are.
Given a random vector $(Y,X)...
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Maximizing the Trace of the Product of Row-Stochastic Matrices
Context: I have found this interesting problem about the trace of row-stochastic matrices and I don't know how to solve it. Thank you in advance for taking the time to look at this probem. I truly ...
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Trace and Norm of an element $\alpha \in \mathbb{F}_{q^{n}}$ is equal to Trace and Determinant of Linear Transform $ T(\beta) = \alpha \beta$
I'm trying to solve the following exercise :
Exercise: Let $\mathbb{{F}_{q^{n}}}$ a extension of $\mathbb{F}_q$ and $\alpha \in \mathbb{{F}_{q^{n}}}$. Define the $\mathbb{F}_q$ - linear map: $T : \...
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$\max \text{Tr}|AXA^T+AYA^T|+\text{Tr}|AXA^T-AYA^T|$ where $X^2=Y^2=I$
Given $A\in\mathbb{R}^{n\times n}$, what is the value for this optimization problem and can this be expressed in terms of $A$?
\begin{align*}
&\max \text{Tr}|AXA^T+AYA^T|+\text{Tr}|AXA^T-AYA^T|...
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$\max{\text{Tr}|AXA^{T}+BXB^{T}|}$ where $X^2=I$
Suppose $A,B\in\mathbb{R}^{n\times n}$ are given matrices. What is the value for this optimization problem and can this be expressed in terms of $A$ and $B$?
\begin{align*}
&\max{\text{Tr}|AXA^...
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Determinant and trace of a linear transformation
Let $V=\mathbb{C}[x,y]$ be the space of polynomials in x,y with complex coefficient. And $H_{n}(x,y)$ be the subspace of homogeneous polynomial of degree n,i.e. $Span_{\mathbb{C}}\{x^iy^{n-i}|0\leq i\...
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Trace of product of some special matrices
Suppose $u^i$, $v^i$ and $w$ are vectors in $\mathbb{R}^n$, with $i=1,2,...,n$. Let $A$ be the matrix with entries $u^i \cdot u^j$, $B$ with entries $v^i \cdot v^j$ and $C$ with entries $w_i w_j$. I ...
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How to show the trace distance is a metric on the space of positive semidefinite matrices?
Let $\mathbb{S}^d_{+}$ be the space of all the $d \times d$ symmetric positive semidefinite matrices. For any $A, B \in \mathbb{S}^d_{+}$, we define
$$d (A, B) = \left[\mathrm{tr}\left(A + B - 2 (A^{1/...
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$ \operatorname{tr}(AA^T)\geq\operatorname{tr}(A^2) $, $ \text{tr}(AA^TAA^T)\geq\text{tr}(A^4) $, and $ \text{tr}((AA^T)^n)\geq\text{tr}(A^{2n}) $ [duplicate]
My friend discovered a different approch of this kind of problem such as:
If A is a real matrix such that $(AA^T)^n=A^{2n}$ or $(AA^T)^nA=A^{2n+1}$, then $A^n=(A^T)^n.$
by making use of trace as ...
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Matrix derivation for $\frac{\partial Tr(AX^2)}{\partial X}$
Suppose matrix $A,X$ are dimension-$n$ matrices. We have $Tr(AX^2)=\sum_{ijk}a_{ij}x_{jk}x_{ki}$. Can we express $\frac{\partial Tr(AX^2)}{\partial X}$ in a simple way, that is in a matrix form ...
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How come $\operatorname{Tr}(AB) = \operatorname{Tr}(BA)$ in general, when $\operatorname{Tr}([x,p_x]) = i\hbar$
If we define $[A,B] = AB-BA$, we know that for any couple $(A,B)$ of $n\times n$ matrices with complex entries, then $\operatorname{Tr}(AB)=\operatorname{Tr}(BA)$ holds.
The latter holds regardless of ...
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Pairwise orthogonality of left and right singular vectors of traceless matrices
Let us consider complex, square traceless matrix $A$. We know that it is unitarily similar to matrix which have 0 on its main diagonal, that is for $B = QAQ^H, b_{ii} = 0$.
Now, let us consider ...
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Proof for Inner Product for a Special Case
If we have an inner product defined as $ \langle A,B \rangle = Tr(A ^\dagger B)$ , how do i prove that this is indeed an inner product for linear operators $A$ and $B$. Generally I know how to prove ...
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Prove there exist matrices $S$ and $T$ such that $B=S(A+T)S^{-1} -T$ iff $\text{tr}(A)=\text{tr}(B)$.
So this problem appeared on a list called "Matrix Problems", meant to be like training for competition-level maths. The whole problem goes like this: Let $A$ and $B$ be $n\times n$ matrices ...
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Find the trace of the matrix multiplied $n$ times [duplicate]
Let $A=[a_{ij}]_{n\times n}$ such that $a_{ij}=\dfrac{(-1)^i(2i^2+1)}{4j^4+1}$. Then, the value of $1+\lim\limits_{n\rightarrow \infty} (\text{Trace}(A^n))^{\frac{1}{n}}$ is ______
The trace of the ...
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Proof of Trace Inequality with Nilpotent Matrices?
I have a conjecture on a trace inequality about a positive-definite, symmetric matrix $D\in\mathbb{R}^{M\times M}$ and a 'conveyor belt' matrix $P\in\mathbb{R}^{M\times M}$:
\begin{equation}
P = \...
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Is there a "clean" proof to $tr(\sqrt{A^* A}) \geq |tr(A)|$?
I'm aware this is a specific case to the general Von Neumann's trace inequality and I am familiar to some ways to prove it, but I was wondering if there would be a proof for this that required not as ...
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Find the trace of matrix be its minimal polynomial
I found this problem in last year materials of my linear algebra course. And I am really interested in solving it
There is a Matrix
$$\begin{matrix}
3 & * & * & * & * & 9 \\
* &...
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Is there any analytical expression for $\mathbf A\,\vdots\,\mathbf B$ as there is for $\mathbf A : \mathbf B=\mathrm{Tr}(\mathbf A\cdot\mathbf B)$?
We know the contraction product or double-dot product of two matrices is
$$\mathbf A:\mathbf B=A_{ij}B_{ji}=(A_{ij}B_{jk})_{i=k}=\mathrm{Tr}(\mathbf A\cdot\mathbf B)$$
The next case that might occur ...
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Identities for power of quadratic forms
Let a quadratic form $q=x^TAx$ where $x\in\mathbb{R}^{n\times 1}$ and $A\in\mathbb{R}^{n\times n}$. I want to express $q^k$ in such a way that the terms that depend on $A$ can be separated (in the ...
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Prove: Derivative of a Trace is the Trace of a Derivative
For some matrix $X$, I am trying to prove that:
$$ \frac{d}{d\theta} \text{Tr}(X) = \text{Tr}\left(\frac{d}{d\theta}X\right) $$
Here is my logic:
For an $n×n$ matrix $X$:
$$ \text{Tr}(X) = \sum_{i=1}^...
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$V(X^\top AX)=tr(\Sigma A\Sigma A) + tr(\Sigma A\Sigma A^\top)$
Let $X$ be an $n$-dimensional random vector with $E[X]=0$ and $E[XX^\top] = \Sigma$, and let $A$ be a matrix in $\mathbb R^{n\times n}$.
A paper I'm reading claims that $V(X^\top A X)=tr(\Sigma A\...
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Trace class condition equivalence
Operators $A$ and $B$ on an infinite Hilbert space are bounded and invertible. I am trying to prove a conjecture for $A,B$ such that
\begin{equation}\tag{i}
(A-I)(B-I),(B-I)(A-I) \in \mathcal{L}_1,
\...
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Weak* convergence in trace class
Given $A_n \overset{*}{\rightharpoonup} A$ in trace class and $\mathrm{Trace}\left|A_n\right| = 1$. I want to prove/disprove that $\mathrm{Trace}\left|A_n - A\right| + \mathrm{Trace}\left|A\right| \to ...
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Extending of Sobolev function in the domain with non-smooth boundaries
I have a function $u(x, t) \in W^{2, 1}_q$ (2 spatial, 1 time Sobolev derivatives in $L_q$ space) with $q>3$ defined in a parabolic cilynder $x \in (-1, 1)$, $t > 0$. Note that Sobolev embedding ...
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Computing the trace of the product $\prod_{k = 1}^{2018}(kA + I_2)$
I have the matrix
$A = \begin{pmatrix} 3 & -6\\1 & -2\end{pmatrix}$
and I want to compute the trace of the product
$$\prod_{k = 1}^{2018}(kA + I_2).$$
I am not sure what to do about this. I ...
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Bounding the Determinant of a Matrix by its Frobenius norm?
I'm almost sure I've done something wrong, and could use some input about it if possible.
Let $A$ be a $n\times n$ square matrix with positive real entries.
Let $B = A^T A$, so that $B$ is positive ...
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Trace norm and operator norm the largest and smallest unitarily invariant norms?
A unitarily invariant norm $\|\ \|_i$ is a norm on operators on a Hilbert space satisfying $\|A\|_i=\|UAV\|_i$, where $U,V$ are unitary and $A$ is any operator. Impose the normalization that $\|P\|_i=...
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Is Trace operation commutative when weighted by a positive semidefinite matrix?
The commutativity of Trace operation is often discussed to be a consequence of the matrix product. One writes
$$tr(AB) = tr(BA) \qquad (1)$$
Now consider the situation where given a positive ...
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Sobolev-type spaces and orthonormal systems
Let $\mathcal{O}$ be a bounded domain in $\mathbb{R}^d$ with smooth boundary. Suppose that $\{e_n\}$ is an orthonormal system in $\mathbb{L}^2(\mathcal{O})$ and let $(\lambda_n)$ be a sequence with $\...
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Optimization of Objective Functions Involving the Sum of Squares of Diagonal Elements in Matrices
How can I optimize an objective function that includes the sum of the squares of diagonal elements? For example, for the optimization problem
$$\text{tr}((A^TB - I)(A^TB - I)^T) - \sum_{i=1}^n (A^TB)_{...
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How to calculate the trace of this matrix?
How do we calculate the trace of $(\mathbb{I}-X(X^\prime X)^{-1}X^\prime)\mathbb{J}_n$? This question stemmed from the below problem I came across:
Suppose we have the linear regression model: $y_i=...
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Evaluating a Difference of Trace Powers of Self-Adjoint Bounded Operators using Only the Spectrum
Let $\mathcal{H}$ be a separable infinite-dimensional Hilbert Space. Let $A$ be a bounded, self-adjoint operator on $\mathcal{H}$ with purely continuous spectrum and let $B$ be a trace-class, self-...
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Maximization of the trace with respect to a permutation matrix
Suppose there are given two real symmetric matrices $\pmb A$ and $\pmb B$. I want to find a permutation matrix $\pmb P$ such that
$\mathrm{Tr}(\pmb P^T\pmb A\pmb P\pmb B) \to \max$
My current status ...
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An equivalent form for the trace-class norm
Let $H$ be Hilbert space,
Let $\mathscr{B}(H)$,$\mathscr{B}_0(H)$, $\mathscr{B}_1(H)$ be , respectivetly ,the bounded, compact and trace-class operator space,
Let define $\| A\|_1 = tr(|A|)$,
I ...
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Applying the Cyclic Property of the Trace for Non-Square Vectors
I know that:
$$x^T A x = \text{trace}(x^T A x)$$
Since $x^T A x $ is a scalar, its trace is itself. However, why is it permissible to use the cyclic property to rearrange this expression if x is not ...
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Can the trace be computed in any Schauder basis?
Edit: This question didn't get any answers after a bounty, so I cross-posted it here on Math Overflow.
Let $H$ be a separable Hilbert space and $T \in L(H)$ a trace-class operator. It is well known ...
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Compact + absolute convergence of eigenvalues $\Rightarrow$ trace class?
Let $H$ be a complex Hilbert space. Lidskii’s theorem says that if an operator $T \in B(H)$ is trace class, then $\operatorname{tr}(T) = \sum_i \lambda_i$, where the sum includes all nonzero ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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$f=0$ 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 ...
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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 $...
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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 ...
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