All Questions
Tagged with fa.functional-analysis linear-algebra
325 questions
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Extending linear operators to multi-linear ones
Suppose we are given a linear operator $L$ on a Banach space $X$. Is there any way to extend $L$ to a multi-linear operator $\mathcal{L}$ in such a way that
$$\mathcal{L}(x_1, x_2^*, \ldots, x_n^*) = ...
user7807
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0
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125
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Transforming nilpotency into diagonalizability [closed]
We designate the $k$-th standard vector as $e_k$ in $\mathbb{C}^n$.
We consider the backward shift operator, denoted as $T: \mathbb{C}^n \to \mathbb{C}^n$, which is defined as follows:
$Te_1=0$ and $...
- 4,058
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44
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Characterizing some similarity invariant homogeneous log-superharmonic functions of matrices
Let $L:M_n(\mathbb{C})^r\rightarrow[0,\infty)$ be a function that satisfies the following properties:
$\log(L)$ is plurisubharmonic.
$L$ is homogeneous in the sense that $L(\lambda A_1,\dots,\lambda ...
1
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133
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Infinite dimensional matrix solvability
In order to solve a boundary problem for a conducting hemisphere, the following matrix equation arises (derived from the boundary condition on the curved part of the surface) where we must solve for ...
- 573
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87
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Continuous choice of null directions for a family of bilinear forms
Let $E$ and $F$ be (real) Hilbert spaces, where $\dim E = \infty$ and $1 \leq \dim F < \infty$. Let $T : E \to \operatorname{Sym}(F \times F,\mathbb{R})$ be a continuous linear map, where $\...
- 2,095
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64
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Small perturbation to a commuting family of hermitian matrices will hurt the nice properties?
Let $A_1, \dotsc A_N$ be a collection of finite Hermitian matrices that commute with one another and all have the matrix $2$-norm as $1$. Here $N$ is large but fixed.
Then, they are simultaneously ...
- 3,477
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72
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Weak majorizations for sum of two hermitian matrices
Let $A$ and $B$ be two $n\times n$ hermitian matrices. Does $U^{*}AU+B \prec_{w} A+B$ for any unitary matrix $U$? Here the notation $``\prec_{w}"$ stands for the weak majorization, that is, $x\prec_{...
- 45
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87
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Computation of the trace of a convolution operator
I try to understand a particular step on page two from the research paper "A multidimensional analog of a limit theorem of G. Szegö".
https://iopscience.iop.org/article/10.1070/...
- 197
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61
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Discrete Orlicz space estimate
We consider the discrete LlogL space of sequences $x=(x_i)$ such that
$$\Vert x\Vert_{LL}:=\sum_i \vert x_i \log(x_i)\vert <\infty.$$
Let $x=(x_i)$ and $y=(y_i)$ two sequences in the above LlogL ...
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70
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Lower-bounding the eigenvalues of a certain positive-semidefinite kernel matrix, as a function of the norm of the input matrix
Let $\phi:[-1,1] \to \mathbb R$ be a function such that
$\phi$ is $\mathcal C^\infty$ on $(-1,1)$.
$\phi$ is continuous at $\pm 1$.
For concreteness, and if it helps, In my specific problem I have $\...
- 6,853
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448
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Smallest eigenvalue for large kernel matrix
I am interested in the the asymptotics of the minimum eigenvalue $\lambda_n^n$ of a class of kernel matrix $P = [ K(x_i - x_j) ]_{i,j}$, with $x_i$ equally spaced in the unit cube of $\mathbb{R}^d$.
...
- 151
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Is there an easy characterisation (perhaps some generalised Löwner representation) for operator monotone functions of order $n$?
As per my understanding, roughly stated, $f$ is an operator monotone function of order $n$ if for all $n\times n$ (Hermitian) matrices, $X,Y\ge0$ which satisfy $X\ge Y$, we have $f(X)\ge f(Y)$.
If $f$...
- 131
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54
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When does the multi-spectral radius coincide with the spectral radius of the sum of linear transformations?
Suppose that $X$ is a finite dimensional Hilbert space and
$A_{1},\dots,A_{r}:X\rightarrow X$ are linear transformations. Define the multi-spectral radius $\rho(A_{1},\dots,A_{r})$ to be
$$\limsup_{n\...
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127
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What is the analogue of expansive matrix for automorphisms?
We say an invertible $n \times n$ matrix with entries in $\Bbb R^n$ is expansive if the absolute values of all of its eigenvalues exceed $1$. An easy calculation also shows that if we consider a ball ...
- 41
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94
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Space spanned by pointwise squares of basis functions
Consider the Hilbert space $L^2(\Omega)$ over some Euclidean domain $\Omega$.
Let $F=\{f_i;i\in\mathbb N\}$ be an orthonormal basis of this space consisting of functions in $L^2(\Omega)\cap L^4(\Omega)...
- 8,086
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256
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Significance of Tikhonov matrix
I am looking for a tutorial on Tikhonov matrix, in the sense what it can do or it cannot do. The definition of the matrix can be obtained in the wikipedia link. https://en.wikipedia.org/wiki/...
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83
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What is optimal distance between inverse of convolution operator?
I am looking for a measure to find the optimal distance measure between inverse of an convolution operator $A$ and say another convolution operator $B$. I want my measure to be sharp that mean when $B$...
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174
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Negative eigenvalue of Toeplitz Hermitian matrix?
I am working on estimation of a covariance matrix and I know that the matrix is Toeplitz. The desired matrix should not produce negative eigenvalues at all. However, sometime my estimation leads to a ...
- 495
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123
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Can we have extension of Mercer theorem to interpolation? [closed]
This question is related to Mercer theorem, Reproducible kernel Hilbert space(RKHS) and interpolation. The wikipedia links are https://en.wikipedia.org/wiki/Mercer%27s_theorem and https://en.wikipedia....
- 11
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85
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What are good bounds on ratios of subdeterminants?
Let $A$ be a symmetric matrix and $A_i$ be the matrix obtained from $A$ by dropping the $i^{th}$ row and column. Then what are some good bounds on the value of $\frac{det(A_i)}{det(A)}$ ?
Using the ...
- 1,893
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270
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Eigenvalue of product of self adjoint compact operators
Suppose A is a self adjoint $m \times m$ real matrix with eigenpairs $\{e_j, \lambda_j\}$ such that $\lambda_j > \lambda_{j + 1}$. Let $B$ be another self adjoint real $m \times m$ matrix such that ...
- 157
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147
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Bounding Rayleigh quotient for stochastic matrix
Suppose you have an irreducible, stochastic matrix $A$ with left Perron-Frobenius eigenvector $v$ (corresponding to the eigenvalue $1$), and suppose the next largest eigenvalue for $A$ is $\lambda$. ...
- 121
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174
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Eigenvalues of a Parametrized Family of Linear Functions
Suppose that we have a family of linear functions $L(\alpha) : \mathbb{R}^n \rightarrow \mathbb{R}^n$, where $\alpha$ is a positive real number.
For each $\alpha$, it is given that $L(\alpha)$ is a ...
- 111
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466
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Bounding point-wise maximum of the absolute difference of two convex functions
Let $\Delta: R \times R \rightarrow R_{+}$ be a positive and convex function (convex in, say, both the arguments) called the loss function.
Let $x \in R^d$. Moreover, let $H_1,...,H_r$ be sets of ...
- 11
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177
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$Ax=b$ in a function space
Let
$X$ be compact Hausdorff topological space,
$C(X)$ denote the algebra of complex-valued continuous functions on $X$,
$b\in \mathbb{C}^m$,
$\mathbf{A}\in C(X)^{m\times n}$,
for all $x\in X$, $b\...
- 85
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228
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Norm equivalence in finite dimensions - is the equivalence "universal" if the dimension is fixed?
I am aware that in a finite dimensional vector space, any two norms are equivalent.
However, I cannot really figure out how "universal" the equivalence constants are.
To be specific, let us ...
- 3,477
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317
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A variation of the Riesz Lemma
Given a normed space $X$, a closed proper subspace $Y$ and $\alpha\in (0,1)$, the Riesz Lemma states that there is $x\in X$ such that $\|x\|=1$ and $d(x,Y)>\alpha$. Observe that also $d(-x,Y)=d(x,Y)...
- 153
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1
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276
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Positive definite Hermitian matrices of countable rank
Say that a $\omega\times \omega$ Hermitian matrix $A$ is positive semidefinite of rank $n$ if there exists a $\omega\times n$ complex matrix $B$ such that $A=B B^\dagger$ where $^\dagger$ denotes the ...
user2529
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114
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Geometric interpretation of a Grammian-like function
Let $\mathbf{v}, \mathbf{w} \in \mathbb{R}^n$ and consider the following function $f : \mathbb{R}^n \times \mathbb{R}^n \rightarrow \mathbb{R}$:
$$
f(\mathbf{v},\mathbf{w}) = \|\mathbf{v}\|\|\mathbf{w}...
- 6,351
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100
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Projection on a countable union of linear subspace
For any natural number $n$, $V_n$ denotes a closed linear subspace of a $L_2(m)$ space, which is an Hilbert Space, where $m$ denotes a finite measure. Moreover $(V_n)$ is increasing, that is $V_n$ is ...
- 31
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269
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Limit of eigenvalues of a matrix perturbation sequence
Suppose $H$ is an $n\times n$ symmetric positive definite matrix, $M_k$ is a sequence of $n \times n$ matrix (not necessarily symmetric) such that $M_k \to O$ where $O$ is the zero matrix. Let $\...
- 135
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317
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Some questions related to the unitary operators
A unitary operator is a surjective linear operator between complex inner product spaces, which preserves the inner product.
What is the name of the analogue for the real case? Orthogonal operator ...
- 5,529
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143
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Differential form of the multidimensional "orthogonal dilation" operator
For a one-dimensional $f(x)$, the dilation operator $f(ax)$ can be expressed as $\exp(g(D))f(x)$, where $g$ is a closed-form function. This is easily checked by e.g. formal Taylor series expansion.
...
- 286
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75
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The spectrum of the product $JA$ where $J=I_n\oplus (-I_n)$
Let $A$ be a real symmetric matrix in $M_{2n}(\mathbb{R})$with $A^2=I_{2n}$. Suppose that the Schur decomposition of $A$ is given by $A=\Lambda^t D \Lambda$. Let us consider the following matrix.
$$...
- 4,058
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1
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110
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Number theory for operator bound
Let $\gamma_i$ be such that for even $i$ $\gamma_i=1$ and for odd $i$ $\gamma_i$ shall have absolute value $1$ and the product of all of the odd ones is also on the complex unit circle but not 1 or -1....
- 501
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1
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198
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The eigenfunctions of an operator commuting with all rotations.
When reading the paper
E. Carlen, J. Geronimo & M. Loss: SIAM J. MATH. ANAL., vol. 40, no. 1, 327-374
I found an argument like the following.
Given an bounded and self-adjoint linear operator ...
- 199
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371
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Is there relation between vector valued RKHS and interpolation space?
Vector valued RKHS which is covered extensively in the book "Pick Interpolation and Hilbert function spaces" . In a different context interpolation space is defined in the wikipedia link: https://en....
- 495
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2
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415
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Commutative *-subrings of the noncommutative C*-algebra $B(l^2)$
A $\star$-ring is a ring with an involutive anti-automorphism. The simplest example of a noncommutative $\star$-ring is perhaps $B(l^2)$, the ring of bounded linear functions on the sequence space $l^...
user2529
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263
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Separability of inner product to a product of Minkowski function and norm
I’ve encountered the following assumption:
Let D be a set such that there exists a Minkowski function $f(u)$ on $\mathbb{R}^l$ and norm $g(v)$ on $\mathbb{R}^m$ such that
$\forall u\in \mathbb{R}^l, \...
- 1
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0
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66
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Taking trace of a tensor product of matrix-valued smooth functions on the thin diagonal
Let $V$ be a finite dimensional real / complex vector space and consider the space $L(V,V)$ of linear operators on $V$.
Fix $n \in \mathbb{N}$ and let $\mathcal{M}$ be the real / complex vector space ...
- 3,477
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46
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What's the problem in using spanning Bessel sequences that are not frames to decompose vectors?
This is related to a question I recently asked on math.SE.
Consider a subset $G\equiv \{g_k\}_{k\in\mathbb{N} }\subseteq\mathcal H$ in a separable Hilbert space $\mathcal H$, and suppose $G$ spans the ...
- 342
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93
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Orthogonalization of symmetric non-degenerate bilinear forms
It is well-known that given a field $k$ with characteristic different from $2$, every symmetric non-degenerate bilinear form $B$ over a finite-dimensional space can be orthogonalized. This means that ...
- 1,485
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55
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Johnson-Lindenstrauss type result for matrix factorization
The type of result I want is: given matrix $A\in \mathbb{R}^{m\times n}$ and error tolerance $\epsilon$, what is a lower bound on $k$ such that $\|A - UV\|_{??}\le \epsilon$, where $U \in\mathbb{R}^{m\...
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0
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64
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When is a symmetric block Toeplitz matrix invertible?
Let
$$
Q =
\begin{bmatrix}
Q_0 & Q_1 & Q_2 & \cdots\\
Q_{-1} & Q_{0} & Q_1 & \cdots\\
Q_{-2} & Q_{-1} & Q_0 & \cdots\\
\vdots & \vdots & \vdots & \ddots
...
0
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0
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163
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Generalization of polynomial coefficients
I'm dealing with a hard combinatorial problem where for every positive integer value of a variable $n$ I have to calculate a list of numbers, specifically $n^2$, that depend on $n$ and its list index ...
- 47
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0
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74
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Computing the eigenvalues of $A+E$ where $A$ is an upper triangular matrix whose diagonal entries are all zero and $E$ is a rank one matrix
Let us consider the backward-shift matrix $B=(b_{ij})\in M_n(\mathbb{R})$ whose entries are given by $b_{k,k+1}=1$ and the other entries are all 0. We also consider $X=(x_{ij})\in M_n(\mathbb{R})$ ...
- 4,058
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0
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146
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Linear dynamics in a function space
I posted the same question to Math Stackexchange earlier without much luck, so I am posting here.
I am dealing with a time-dependent model, which can be expressed as a function. $f$ is dependent on ...
- 433
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0
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124
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Eigenvectors of the symmetric tridiagonal matrices whose entries above the diagonal are all the same
Let us consider the real symmetric tridiagonal matrix $T=(t_{kl})$ in $M_n(\mathbb{R})$ with
$$t_{1,2}=t_{2,3}=\cdots=t_{n-1,n}=1$$
How can we compute the eigenvectors of $T$?
- 4,058
0
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0
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272
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Finding the eigenvectors of a submatrix
Let $A=(a_{kl})$ be a matrix in $M_n(\mathbb{R})$ when $n$ is even. Let $B=(b_{kl})$ be the symmetric $2n$ by $2n$ matrix whose entries are given by,
$b_{k,l}=a_{kl}$ if $1\leq k,l\leq n$.
$b_{n+k,l}=...
- 4,058
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0
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104
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Necessary and sufficient conditions for $\mbox{trace}(A^{-1/2}e^{-tB} (AB+BA) e^{-tB}A^{-1/2}) \ge 0$ for all $t$
Let $A$ and $B$ be positive-definite matrices of the same size. For any $t \ge 0$, define
$$
u(t) := \mbox{trace}(A^{-1/2}e^{-tB} (AB+BA) e^{-tB}A^{-1/2}).
$$
Question. What are necessary and ...
- 6,853