All Questions
Tagged with operator-theory matrices
319
questions
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40
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A Question about the Matrix Representaition of an Operator
I'm reading an argument which assumes that $H$ is a Hilbert space and $A\in B(H)$.I'm having difficulty understanding one matrix representaion of $A$.
First, the author decomposes $H$ as $H=\overline{...
- 573
2
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1
answer
42
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Equivalence of Kraus operators on a single element
Let's say that I'm working in a Hilbert space $\mathcal{H}$,suppose I have two bounded operators $A.B\in B(\mathcal{H})$ and a positive semidefinite operator $x$, for example take $x$ to be a density ...
- 75
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41
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Is every linear relation on $\mathbb{C}^2\times \mathbb{C}^2$ represented by two matrices?
Let us consider the following definition.
Definition. Let $H_1,H_2$ be Hilbert spaces. A linear subspace $D$ of $H_1\oplus H_2$ is called a linear relation from $H_1$ to $H_2$.
Now, let $H_1=H_2=\...
- 355
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23
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$\|B\|_{op} \le \|A\|_{op}$ when $B_{i,j} = v_i^TAv_j$
I'm currently stuck to the following statement.
Let $A\in\mathbb R^{n\times n}$ be a diagonal matrix whose diagonal elements are only 0 or 1, and let $B$ be $n\times n$ matrix such that $B_{i,j} = ...
- 769
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When is $\Vert A+B \Vert = \Vert X + X^\dagger \Vert$ for $[A, X; X^\dagger, B] \succcurlyeq 0$?
Given
$$
H = \begin{pmatrix} A & X \\
X^\dagger & B
\end{pmatrix}
\succcurlyeq 0 ,
$$
with $A \succcurlyeq 0$ meaning $A$ is positive semi-definite (and Hermitian)
and $A, B, X$ are $n\...
- 3,560
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36
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Bounded (finite-rank) Inverse Operators
While studying for my functional analysis course I encountered bounded inverse theorem which states that given a bijective bounded linear operator $T:X\rightarrow Y$, then $T^{-1}:Y\rightarrow X$ is ...
- 35
1
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1
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83
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Inequality regarding Matrix Norm and Inverse Matrix
Currently, I'm stuck to one of a statement in a paper. Following is a brief summary of the paper regarding my question. (although the topic of the paper is mainly statistics, the question purely ...
- 769
1
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0
answers
51
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Show that $\Vert M \Vert \le 1/\lambda$ for $M = (S+\lambda I + \lambda\eta A)^{-1}$
For the given matrix
$$M = (S+\lambda I + \lambda\eta A)^{-1},$$
where $S,A$ are positive definite matrix, $I$ is identity matrix, $\lambda, \eta > 0$, my goal is to show that $\Vert M \Vert \le 1/\...
- 769
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0
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21
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Criterion for a non-trivial overlap of null spaces of real-symmetric matrices
I have two large real-symmetric matrices $A,B\in \mathbb{R}^{N\times N}$, which obey $A \cdot B = 0 = B \cdot A$, meaning that they both commute (and thus share eigenvectors) and anticommute.
I am ...
- 576
1
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0
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36
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Matrix-order derivatives (differentiating a function a matrix number of times)
I have been exploring methods of generalizing the order of derivatives to a broader range of inputs (such as real numbers, complex, and now matrices). We are very well familiar with integer-order ...
1
vote
2
answers
96
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Let $M \in \mathbb{R}^{d \times d}$ and $k \in \mathbb{N}$ prove that $f(M)=M^k$ is differentiable.
Question:
Let $M \in \mathbb{R}^{d \times d}$ and $k \in \mathbb{N}$ prove that $f(M)=M^k$ is differentiable.
My Answer:
In order to prove that $f(M)=M^k$ is differentiable we have to show by ...
- 805
0
votes
1
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93
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How to diagonalize an infinte dimensional operator
I want to take logarithm of an infinite dimensional operator given by
$\rho = \int\int dx_1 dx_1'C(x_1,x_2)C^*(x_1',x_2)|x_1\rangle\langle x_1'
|$, where $C(x_1,x_2)$ is a gaussian function in $x_1$ ...
1
vote
1
answer
50
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Extending the matrix property to nilpotent operators in infinite dimensional spaces. [closed]
Let $A \in M_n(\mathbb{C})$ be nilpotent matrix such that $A^{4} = 0$, We can find matrix $B \in M_n(\mathbb{C})$ such that we have $$AB \neq 0 \ \ and \ \ BA = 0$$
The question is as follows:
How can ...
0
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1
answer
149
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A Property of the Trace-Norm
Let, $\mathcal{E}$ be a Completely-Positive Trace-Preserving Map, i.e, for linear operators $\rho$ on a Hilbert-Space,
$$
\mathcal{E}(\rho) = \sum_i E_i\rho E_i^{\dagger}\qquad {\rm s.t.}\qquad \sum_i ...
1
vote
1
answer
43
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Quaternionic numerical range of a class of matrices
I need to characterize the quaternionic numerical range of matrices of the form $$\begin{bmatrix} r & p \\ 0 & -r\end{bmatrix}$$ where $r$ is a positive real number and $p$ is a quaternion. ...
