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.

Filter by
Sorted by
Tagged with
-2 votes
0 answers
9 views

If $\operatorname{trace}((cI+A)^{-3}A(cB-2I))<0$, is it true that $\operatorname{trace}((cI+A)^{-4}A(cB-2I))<0$?

Given a nonnegative scalar $c$, two positive definite matrices $A$ and $B$, if $\operatorname{trace}((cI+A)^{-3}A(cB-2I))<0$, is it true that $\operatorname{trace}((cI+A)^{-4}A(cB-2I))<0$?
user avatar
0 votes
0 answers
9 views

Non-scalar quadratic forms with non-PSD and non-Hermitian coefficient matrices

I need to implement the expression $obj = \mbox{tr}(AXB'Y'A'X'BY)$ in CVX, where $A$ and $B$ are known complex matrices and $X$ and $Y$ are unknown complex matrices, which I intend to optimize $X$ ...
user avatar
  • 1
1 vote
0 answers
27 views

Is the trace of this operator finite? [duplicate]

Let $H$ Hilbert space and let $Q\colon H \to H$ be a linear, self-adjoint, positive, trace-class operator. Let $X\colon H \to H$ be a linear, self-adjoint, positive operator. Does it follow then that $...
user avatar
0 votes
1 answer
24 views

Derivative of Frobenius norm with respect to scalar

Can anyone explain to me why $$\frac{d}{d\theta }\left\|\textbf{Aw}-\theta(1-c)\textbf{1} \right\|^{2}= 0$$ is equivalent to $$2(1-c)\textbf{1}^T(\theta(1-c)\textbf{1}-\textbf{Aw})=0$$? I'm not very ...
user avatar
  • 11
2 votes
0 answers
54 views

Can we write down $\mathrm{vec}(Y^{-1})$ in terms of $\mathrm{vec}(Y)$?

Lets denote the Kronecker product by $\otimes$ and the vectorization of a matrix $Y$ by $\mathrm{vec}(Y)$. Given $A\in\mathbb{R}^{n\times n}$ and $B\in\mathbb{R}^{n\times m}$, where $n\geq m$. What ...
user avatar
  • 1,559
0 votes
1 answer
20 views

Bound on the trace of a product of a symmetric positive definite matrix and a nil-potent matrix.

Let $A$ a matrix with the following form: $A = vu^{T}$ where $u$ and $v$ are orthogonal vectors and $B$ be a symmetric positive definite matrix. Is there a way to upper bound $Trace(AB)$? I know that ...
user avatar
  • 13
4 votes
1 answer
167 views

Minimize $\mathrm{tr}(B^TXB)$ subject to $X=A^TX(I+BB^TX)^{-1}A$

For a given $A=\begin{bmatrix}a_1&&&\\&a_2&1&\\&&a_2&\\&&&a_2\end{bmatrix}$ and $B=\begin{bmatrix}b_{11}&b_{12}\\b_{21}&b_{22}\\b_{31}&b_{32}...
user avatar
  • 1,559
1 vote
1 answer
17 views

Upper bound on trace for powers of a matrix which converges to zero

For some fixed square matrix $X$ for which $\lim_{n\rightarrow\infty} X^n = 0$, it is apparent that the trace of $X^n$ must also converge to zero as $n\rightarrow\infty$. Is there anything I can do to ...
user avatar
0 votes
1 answer
26 views

Trace of a power of a skew-symmetric matrix

How to express ${\rm Tr}(A^n)$ (in terms of ${\rm det}\,A$), where $A$ is a skew-symmetric $m\times m$ matrix? With references if possible.
user avatar
  • 257
1 vote
1 answer
29 views

Cross covariance and trace identity

This may be a simple answer but I can't find any proof. I know that the following identities are true $$ E\left \{ \left ( \mathbf{x} - E\left ( \mathbf{x} \right ) \right )^T\mathbf{Q}\left ( \...
user avatar
1 vote
0 answers
30 views

When is the trace of a matrix group surjective over $\mathbb{F}_p$?

Let $p$ be a prime and $G\subset\operatorname{GL}_n(\mathbb{F}_p)$ be a subgroup. I wondering about the following question: Is the map $\operatorname{Tr}:G\to\mathbb{F}_p$ surjective? I know it's true ...
user avatar
  • 187
0 votes
1 answer
46 views

Is it possible to produce identically-behaving binary extension fields using different irreducible polynomials?

Let $GF(2^m)$ be a binary extension field with constructing polynomial $f(z)$ be an irreducible, primitive polynomial over $GF(2)$. Is there any possibility that two (or more) different $f(z)$ can ...
user avatar
1 vote
1 answer
57 views

Trace and Lie derivative of a $(1,1)$-tensor commute (Direct proof)

My question is same as this MSE post but I want to use direct properties of Lie derivative and trace to prove (I know another proof using this fact that pullback map commutes with contraction) $$\...
user avatar
  • 7,829
0 votes
0 answers
27 views

Simplification of a Partial Trace

After some derivations, I arrived at the following result. Given the following matrices $\mathbf{C}_{ij}\in R^{n\times n}$, $i,j\in[1..m]$ and the diagonal matrix $\mathbf{Q}\in R^{n\times n}$, I have ...
user avatar
1 vote
0 answers
21 views

Projection of tensor in vectorial space

Let us consider the closed convex cone of Sym (Sym be the subspaces of Lin constituted by all symmetric tensors) K={A∈ Sym such that trA ≤ 0 } For each E∈ Sym determine the projection P(E) of E onto K
user avatar
0 votes
1 answer
32 views

trace Inner product

Attempt I have come across this sequence of questions whilst revising for my linear algebra exam,I have tried to answer these questions (with my attempt attached) but I am in no way sure that I have ...
user avatar
3 votes
2 answers
68 views

Multiplicative property of trace

Let $T_1 \in \mathcal{L}(V)$ and $T_2 \in \mathcal{L}(V)$ be positive operators. Prove that the trace of their product is non-negative i.e., tr($T_1 T_2) \geq 0$ Attempt 1: Obviously, a positive ...
user avatar
0 votes
1 answer
43 views

Trace of Kronecker product $tr((I_N\otimes \Sigma_2)(\Sigma_1\otimes I_N))$

Let $\Sigma_1, \Sigma_2\in\mathbb{R}^{d\times d}$ symmetric and positive definit and $N\in\mathbb{N}$, $N\neq d$. Can the following term be simplified? $$tr((I_N\otimes \Sigma_2)(\Sigma_1\otimes I_N))$...
user avatar
0 votes
1 answer
43 views

How to solve the matrix equation $\mbox{tr}(\mathbf{X}\mathbf{X}^H)\overline{\mathbf{B}}=\mbox{tr}(\mathbf{X}\mathbf{X}^H\mathbf{B})\mathbf{I}$?

I want to solve the following equation for $\mathbf{X}\in\mathbb{C}^{N\times M}$, with $M < N$: $$\mbox{tr}(\mathbf{X}\mathbf{X}^H)\overline{\mathbf{B}}=\mbox{tr}(\mathbf{X}\mathbf{X}^H\mathbf{B})\...
user avatar
  • 41
0 votes
0 answers
30 views

Application for the trace inequality.

I know two theorems about the trace inequality. Suppose that $\Omega$ is a bounded domain with smooth boundary. One is that: $$ \gamma_0(H^1(\Omega)) = H^{\frac{1}{2}}(\partial \Omega) $$ where $\ \ ...
user avatar
  • 161
1 vote
1 answer
61 views

derivate and chain rule about trace

suppose $\mathbf{x}\in \mathbb{R}^d $ with size of (1,d), and $\mathbf{T} \in \mathbb{R}^{d \times d}$. Then how to solve the derivate of $\mathbf{x}$: $$\begin{align} \textrm{tr}(\ (\mathbf{x}^{\...
user avatar
  • 23
0 votes
1 answer
20 views

Determine the trace of a operator $M\in L(L(V)), M(S)=S+S^*$

Let $V$ be an n-dimensional real inner product space. Let $M\in L(L(V))$ be defined by $M(S)=S+S^*$. Then determine the trace of $M$ I'm confused by this problem. I think $M$ is an operator $M: L(V)\...
user avatar
  • 754
-3 votes
1 answer
77 views

What is the kernel of a number? [closed]

tr: R^2×2 → R trace \begin{bmatrix}a&b\\c&d\end{bmatrix} Is a linear transformation. What is ker(tr)? I am confused with this question, how can I find the trace of a a real number. All the ...
user avatar
  • 31
0 votes
1 answer
59 views

Relation between $\operatorname{Tr}(AB)$ and $\operatorname{Tr}(A)\operatorname{Tr}(B)$

I assume this is a rather elementary question but does it exist some kind of inequality between $\operatorname{Tr}(AB)$ and $\operatorname{Tr}(A)\operatorname{Tr}(B)$? A priori we do not assume ...
user avatar
  • 857
0 votes
2 answers
91 views

Please help in leading further in: If $\vert A\vert+\vert B\vert =0,$ then find the value of $\vert A+B\vert$. [duplicate]

There are two square matrices $A$ and $B$ of same order such that $A^2=I$ and $B^2=I,$Where $I$ is a unit matrix.If $\vert A\vert+\vert B\vert =0,$ then find the value of $\vert A+B\vert ,$here $\...
user avatar
  • 3,230
1 vote
0 answers
33 views

Invertibility of trace duals of orders in number fields

Let $K \subseteq L$ be number fields and $S$ be an order in $L$ (not necessarily maximal). Let $R:=S\cap K$ and $S^*$ such as $R^*$ denote the trace duals of $S$ and $R$, respectively. Then $S^*$ and $...
user avatar
  • 11
2 votes
1 answer
36 views

Why is $\Phi(X) = \mathrm{Tr}_\mathcal X(J(\Phi)(1_\mathcal Y \otimes X^T))$?

I'm reading about the Choi representation from John Watrous' textbook on quantum information. On page 78, he says that for any choice of complex Euclidean spaces $\mathcal X$ and $\mathcal Y$, one may ...
user avatar
  • 317
2 votes
0 answers
39 views

Compute the rank of an orthgonal projection efficiently

Question. Someone hands you an $n\times n$ matrix and tells you that it is an orthogonal projection. Describe how to compute the rank of this matrix using at most $n$ operations of addition, ...
user avatar
  • 2,228
0 votes
3 answers
122 views

Quadratic form of a trace

Let $V := \left\{ X \in \mathfrak{gl}(2,\mathbb{R}) \mid \mbox{tr}(X) = 0 \right\}$ be a vector space over $\mathbb{R}$. Prove that function $$V\ni X \mapsto q(X) := \mbox{tr}(XDX^{T}),$$ where $$D=\...
user avatar
3 votes
0 answers
67 views

Application of Selberg Pre-Trace formula

I am trying to solve the following problem: Prove the following estimate: $$\sum_{\vert t_i\vert<T}\vert u_j(z)\vert^2\ll T^2$$ by choosing appropriate $h$ in the Selberg's (pre)-trace formula. I ...
user avatar
0 votes
1 answer
55 views

How to show ${\rm tr}[AXA'] \geq {\rm tr}[X]$ for $X\geq 0$ if $AA' \geq I$?

Assume that $A \in \mathbb{R}^{n \times n}$ and $AA' \geq I_n$, where $A'$ is the transpose of $A$, and $X \in \mathbb{R}^{n \times n}$ and $X \geq 0$, i.e., $X$ is a positive matrix . How to show \...
user avatar
  • 405
0 votes
0 answers
91 views

Show that $H \mathrel{\unlhd} SU(2)$ contains matrices with all possible traces between $2$ and $2\cos \varphi_0 $.

Let $H \mathrel{\unlhd} SU(2)$ that contains an element $A$ such that $\text{tr}(A)=2\cos \varphi_0 \neq \pm2$. Use i),ii) to show that $H \mathrel{\unlhd} SU(2)$ contains matrices with all possible ...
user avatar
  • 1,248
-1 votes
1 answer
49 views

Whether ${\rm trace}(\mathbf{x}\mathbf{x}^TM)={\rm trace}(\mathbf{x}^TM\mathbf{x})$ [closed]

If $\mathbf{x} \in R^n,M \in R^{n \times n}$,and trace(.) is the sum of the diagonal elements of a matrix,whether the above conclusion is hold,if so,how to prove,thanks in advance
user avatar
0 votes
0 answers
39 views

Field trace used to show certain elements are not in certain fields or extension of fields

These notes from the Uni. Connecticut state about finite extensions of fields that "among elementary applications, the field trace can be used to show certain numbers are not in certain fields&...
user avatar
0 votes
0 answers
36 views

Calculating Trace of Integral Operators

I could not figure out how the following formula can be derived: $$ \operatorname{Tr} [ W(t) Q(t) ] = \frac{1}{(2\pi)^{d/2}} \int_{\mathbb{R}^d \times \mathbb{R}^d} \widehat{W}(t, q-p) \widehat{Q}(t,p,...
user avatar
0 votes
0 answers
35 views

Trace of the projection operator

Usually the trace of the orthogonal projection onto a finite dimensional space equals to the dimension of the space: Trace$(P)=M$ where $M$ denotes its dimension. For an orthonormal basis $e_1,\cdots,...
user avatar
  • 141
0 votes
0 answers
34 views

Geometric interpretation for norms and traces in number fields or matrix algebras

In a simple algebra $A$ over $\mathbb{Q}$ (one might just think of a number field or a matrix algebra), the trace can be interpreted geometrically as giving an "inner product" for $A$ as a ...
user avatar
  • 195
0 votes
0 answers
32 views

Need help classifying the fixed points of a dynamical system

I am currently studying for an exam on modeling single neuron dynamics and I am stuck at determining whether the fixed points of a dynamical system of two ODEs are spirals or nodes (the FitzHugh-...
user avatar
0 votes
2 answers
61 views

Matrix derivative of the Frobenius norm of a product containing inverse

Let $A\in\mathbb{R^{n\times d}}$, $X\in\mathbb{R^{d\times d}}$, $d>n$. Let $A$ have rank $n$ and let $X$ be invertible. What is the derivative of $$\Vert XA^T(AXA^T)^{-1} - A^T(AA^T)^{-1}\Vert_F^2$$...
user avatar
  • 33
1 vote
1 answer
42 views

Formula for counting the number of subgraphs in a given graph $G$

I want to find a formula for counting the number of subgraphs in a given graph $G$ with adjacency matrix $A$. The subgraph I want to count its number of occurences is of the structure above. So my ...
user avatar
  • 123
-1 votes
1 answer
67 views

Does it hold that $tr(ABC)\le tr (AB)$? [closed]

$A,B$ and $C$ are all real orthogonal projections. So for each $X\in \{A,B,C\}$ it holds that $X^2=X=X^*.$ Does it hold that $$tr(ABC)\le tr (AB)?$$ If no, for what special cases could it hold?
user avatar
  • 75
0 votes
1 answer
43 views

How do you take the derivative of a trace of matrix kroenecker products?

Let $M \in \mathbb{C}^{vr \times vc}$ and $P \in \mathbb{R}^{v \times v}$, where $P \succ 0 $ then, $F = Tr(M (P^{-1} \otimes I_{c}) M^H (P \otimes I_{r}))$, where, $I_{r} \in \mathbb{R}^{r \times r}$ ...
user avatar
6 votes
1 answer
99 views

The trace form is the unique non-deg. symmetric bilinear form such that ...

Let $L \vert K$ be a field extension. Recall the various definitions of the trace map $Tr_{L\vert K}:L\mapsto K$ (via matrices, minimal polynomials, field embeddings). With that, one gets the trace ...
user avatar
0 votes
0 answers
11 views

Express variance of ridge estimator using trace

Consider the model $X = Z b + \epsilon$, where $Var(\epsilon) = \sigma^2 I_n$. $Z$ is an $n \times d$ matrix and $b$ a $d$ dimensional vector. Consider the problem $\min\limits_b \| X-Zb\|^2 + \...
user avatar
  • 345
0 votes
1 answer
25 views

Upper bound for $\mathrm{tr}\Big((P + I)^{-1}(P + d uu^T)(P + I)^{-1}\Big)$

I am interested in the following function, $$ f(P, u) := \mathrm{tr}\Big((P + I)^{-1}(P + d uu^T)(P + I)^{-1}\Big). $$ Above, $P$ is a symmetric positive definite real matrix and $u$ is a vector with ...
user avatar
  • 2,382
1 vote
0 answers
33 views

Find all $6 \times 6$ matrices knowing the trace and $A^3-6A^2+5A=0$

I have to find all matrices $A$ that satisfy: Trace of matrix $A$ is $17$ $A^3-6A^2+5A=0$ I have calculated characteristic polynomial $X(X-1)^2(X-5)^3$ and minimal polynomial $X(X-1)(X-5)$, so ...
user avatar
  • 21
-2 votes
1 answer
40 views

Proof of the following identity [closed]

$\det(I+tA)=1+t\cdot\mathrm{tr}(A)+O(t^2)$ I didn't find a good answer on the details on this for $A$ has only $n\times n$ dimension requirement.
user avatar
  • 47
0 votes
1 answer
41 views

Why $ (\mathop{tr} B)^2 =\sum_{i,j} b_{ii} b_{jj}$? [closed]

Given a symmetric matrix $B$ how we compute $$(\mathop{tr}B)^2 =\sum_{i,j} b_{ii} b_{jj}.$$ Could you please someone cast some light?
user avatar
  • 721
0 votes
1 answer
63 views

Maximizing the trace of $AB(B+A^{-1})^{-1}$.

Given that $A=\text{diag}(a_1,\dots,a_n)$ is a diagonal matrix (this we fix) with positive values and $B=\text{diag}(b_1,\dots,b_n)$ is a another diagonal matrix (this we get to choose) with positive ...
user avatar
  • 395
1 vote
0 answers
25 views

Notation in the context of tensor contraction, partial trace and vector-valued trace

Let $X,Y$ be $\mathbb C$-Hilbert spaces and $X\otimes Y$ denote the completion of the Hilbert space tensor product of $X$ and $Y$. Let $y\in Y$ and $A_y:X\to X\otimes Y\to X$ denote the unique ...
user avatar
  • 12.8k

1
2 3 4 5
31