# Tagged Questions

**14**

votes

**1**answer

250 views

### A possible extension of a determinant inequality

It is well known that if $A, B$ are positive semidefinite matrices, then $$\det (A+B)\ge \det A+\det B.$$
I am considering a possible extension of this result. Let $\mathbb{M}_m(\mathbb{M}_n)$ ...

**5**

votes

**2**answers

311 views

### How to check whether a matrix is completely positive or not?

The definition:
cone of completely positive matrices
$\mathcal{C}=\{\sum_{i=1}^kx_ix_i^T:x_i\in\mathbb{R}^n_+\ for \ i=1,2,...,k\}$.
I just don't knwo how to check whether a matrix belongs to ...

**2**

votes

**2**answers

228 views

### Are the finite dimensional von Neumann algebras, singly generated?

Let $\mathcal{M}$ be a finite dimensional von Neumann algebra, then :
$$\mathcal{M} \simeq \bigoplus_i M_{n_i}(\mathbb{C})$$
Question : Is it singly generated (as von Neumann algebra)? how ?
...

**0**

votes

**0**answers

96 views

### Operator Adjoints and Non-Symmetric Inner Products

Let $V$ be a finite dimensional vector space (over $C$ if that makes a difference), and let $T$ be a linear operator on $V$. Now if $(\cdot,\cdot)$ is an inner product on $V$, then it is well-known ...

**3**

votes

**1**answer

291 views

### What is the significance of matrix ordered algebras?

I am trying to grok matrix ordered operator algebras, but I am having a hard time understanding their significance from the definition. Here is the definition (or at least, one way of stating it):
...

**2**

votes

**0**answers

196 views

### How to determine there exists a unique invariant subspace for a set of matrices

Hi everyone,
Ive been looking at the following problem, but its not entirely in my area and some potential solutions seem to rely on algebraic geometry. Maybe thats just a complicated way to solve ...

**4**

votes

**2**answers

496 views

### When is spectral norm of AB equal to that of BA?

I have $A^{1/2} B A^{1/2} \preceq I$ for two PSD matrices $A$ and $B$, and I'd like to know if that implies $\|AB\|_2 \leq 1.$
The argument I was using to show this is that for any two square ...

**1**

vote

**0**answers

140 views

### Norm bound of the entrywise logarithm of a stochastic matrix stationary matrix

Hello,
Denote $\log_\star$ as the entrywise logarithm operation, and let $A$ be some row-stochastic matrix such that $\lim_{p\rightarrow\infty}A^p$ exists and all its entries are non-zero.
As a part ...

**45**

votes

**2**answers

2k views

### vector balancing problem

I believe the solution posted to the arXiv on June 17 by Marcus, Spielman, and Srivastava is correct.
This problem may be hard, so I don't expect an off-the-cuff solution. But can anyone suggest ...

**1**

vote

**0**answers

127 views

### Diagonalizing matrices of linear forms of indeterminates

Let $B$ be a matrix with elements as linear forms of indeterminates. Is there a proper diagonalization procedure for such matrices like those of matrices with real and complex entries?

**11**

votes

**1**answer

721 views

### Decomposition of positive definite matrices.

It is known that a $n^2 \times n^2$ positive semidefinite matrix $A$ cannot always be written as a finite sum
$$
A=\sum_{j} B_j \otimes C_j
$$
with $B_j$ and $C_j$ positive semidefinite matrices (of ...

**7**

votes

**3**answers

471 views

### Relationship between free probability and deterministic graphs?

Consider the $N\times N$ matrix $$
M = \left(\begin{array} \\
0 & 1 & & 0 \\
1 & \ddots & \ddots & \\
& \ddots & \ddots & 1 \\
0 & & 1 & 0 \\
...

**3**

votes

**1**answer

297 views

### Eigenvalues of certain positive matrices

For a matrix $ Q = (q_{ij}) \in GL_n(\mathbb{C}) $ let
$ \overline{Q} = (\overline{q_{ij}}) $ be the matrix obtained by entry-wise complex
conjugation (equivalently, $ \overline{Q} $ is the ...

**5**

votes

**2**answers

402 views

### Can finitely many hermitian (positive-semidefinite) operators always be embedded into a small dimensional space preserving inner product?

Given $n$ hermitian (positive-semidefinite) operators $Q_1,\cdots,Q_n$ in finite dimensional Hilbert space (the dimension can be very large), is there a mapping $\phi$ maps $Q_i$ to $P_i$, which ...

**4**

votes

**1**answer

628 views

### When is this map completely positive?

Consider the complex $n$-by-$n$ matrices $M_n$.
Suppose that $A_i$, for $i=1,\ldots,n^2$, satisfy $\mathrm{Tr}(A_i^*
A_j)=\delta_{ij}$, so that together they form an orthonormal basis for
$M_n$. ...

**3**

votes

**1**answer

318 views

### Completely bounded maps on Mn

The aim of this question is to collect nice maps on $M_n(\mathbb{C})$ with the following property:
$\phi_n:M_n(\mathbb{C})\rightarrow M_n(\mathbb{C})$ with $||\phi||=1$ and ...

**0**

votes

**3**answers

695 views

### Is there any conclusions generalized Singular Value Decomposition into Hilbert Space

Spectrum decomposition can be regarded as the generalizations of the following fact that:
Every Hermitian matrix $A$ can be decomposed into $A=U^{*}\Lambda U$,where $U$ is a unitary matrix
Singular ...

**16**

votes

**1**answer

1k views

### Commuting unitaries

Is the following true:
For every unit vectors $x_1,..., x_n$, $y_1,..., y_n$ in $\mathbb{C}^k$
there exist a Hilbert space $H$, unitary operators $U_1,...,U_n$ and $V_1,...,V_n$ in $B(H)$ and unit ...

**3**

votes

**0**answers

293 views

### A question about the generalized Lidskii-Wielandt inequality for matrices proved by Thompson and Freede

In 1971, Thomson and Freede generalized the Lidskii-Wielandt inequalites as follows (version for singular values)
Let $A$, $B$ be $n\times n$ Hermitian matrices. Suppose $\alpha_1\geq \alpha_2 \geq ...

**1**

vote

**1**answer

435 views

### Is exp(rA) = (exp(A))^r for real r and A in a Banach space?

Is $e^{(rA)} = (e^{A})^r$ when $r \in \mathbb{R}$ and $A$ is an element of a Banach algebra?
Clearly if $n$ is an integer, then
$e^{(nA)} = e^{A+A \cdots +A} = e^{A}e^{A}\cdots e^{A} = (e^{A})^n$,
...

**0**

votes

**3**answers

262 views

### Existence of tensor product of subalgebras

Let $\mathcal{G} = \mathbb{M}_n(\mathbb{C})$ be an $n$-by-$n$ matrix algebra over complex numbers. Next let $\mathcal{A} \cong \mathbb{M}_d(\mathbb{C})$ be a subalgebra of $\mathcal{G}$ and assume $d$ ...

**3**

votes

**1**answer

261 views

### Bounds on operator 2-norms on partial traces of linearly related operators

Consider an arbitary positive semidefinite operator ρ, acting on ℂA ⊗ ℂB ⊗ ℂC, for A,B,C finite. Also, let P be an orthogonal projector on ...

**1**

vote

**2**answers

143 views

### How to study the behavior of a particular function on a Vector Space.

Let, $V$ be a vector space over a field $K.$ Let, $T$ be a function from $V$ to $V$ such that
$T(kX) = kT(X)$ for all $k \in K$ and for all $X \in V$ and also
$T(k + X) = T(X)$ for all $k \in K$ ...

**2**

votes

**3**answers

523 views

### Are the Gell-Mann matrices extremal when used as Kraus operators for a quantum channel?

Landau and Streater proved that a set of Kraus operators, Ai, is extremal if and only if the set
$\{A_{k}^{\dagger}A_{l}\}_{k,l \ldots N}$
are linearly independent. I have seen very convincing ...