# Tagged Questions

**2**

votes

**0**answers

43 views

### Reducing $\ell_1$ norm of non-full-rank matrices

I have two matrices ${\bf{X}}_{p\times r}$ and ${\bf{Y}}_{r\times q}$ with $r<\min(p,q)$. Matrix ${\bf Y}$ does not have full row rank (i.e., rank$({\bf Y})<r$). Can I build two new matrices ...

**0**

votes

**1**answer

54 views

### Uniqueness of solution of a nonconvex optimization problem

What conditions need to be hold for a nonconvex optimization problem to have a unique solution?
Specifically, I have the following minimization problem that I'd like to know whether it has a unique ...

**0**

votes

**0**answers

35 views

### Invexity of the $L_2$ norm

I have the following function:
$ f({\bf A,b}) = \| {\bf y - XAb} \|_2^2$
where ${\bf y}_{n \times 1}$ and ${\bf X}_{n \times p}$ are fixed, and ${\bf A}_{p \times r}$ and ${\bf b}_{r,1}$ are the ...

**0**

votes

**2**answers

88 views

### Convexity of the Frobenius norm of the product of two matrices

I have the following function for two matrices ${\bf A}$ and ${\bf B}$:
$f({\bf A}, {\bf B}) = \| {\bf Y - XAB} \|_F^2 = trace\{({\bf Y - XAB)}^T({\bf Y - XAB)}\}$
where matrices ${\bf X}_{n \times ...

**2**

votes

**1**answer

81 views

### Approximation with a rank-$1$ matrix

Given a matrix $A$ (generally speaking, complex and non-square), I want to find an identically-sized matrix $D$ with ${\rm rk} D\le 1$ to minimize the induced operator norm $\|A-D\|_2$. From the ...

**1**

vote

**1**answer

77 views

### Bound on the 2-norm of a “special” matrix

Let $S\in\mathbb{R}^{n\times n}$ be such that $\|S\|_2\leq 1$, $P\in\mathbb{R}^{n\times m}$, $m<n$, with orthogonal columns ($P^TP=I$) so that $PP^T$ and $I-PP^T$ are orthogonal projectors, and ...

**5**

votes

**0**answers

245 views

### Symmetric matrices with $\rho(A)\gg\|A\|_\infty$

For a symmetric real matrix $A$, denote by $\rho(A)$ the spectral radius of $A$, and by $\sigma(A)$ the largest absolute row sum of $A$; that is, $\sigma(A)=\max_i \sum_j |a_{ij}|$, where $a_{ij}$ are ...

**2**

votes

**0**answers

150 views

### Norm bound of a complex resolvent

A well known result by Varah states that if $A$ is a strictly diagonally dominant matrix of dimension $n$, then
$\|A^{-1}\|_{\infty} \le \max_i\frac{1}{|a_{kk}|-\sum_{j \neq k}|a_{kj}|}$, where the ...

**5**

votes

**1**answer

117 views

### An inequality involving the spectral norm of a complex matrix

Let $A,B \in {M_n}(R)$ be real $n \times n$ matrices and let matrices $|A|$ and $|B|$ contain the absolute values of the elements of $A$ and $B$ respectively. Construct the complex matrices $C = A + j ...

**10**

votes

**1**answer

302 views

### Hölder's inequality for matrices

I was wondering if the HÃ¶lder's inequality was true for matrix induced norms, i.e. if
$$\|AB\|_1 \leq \|A\|_p\|B\|_q, \quad\forall p,q \in [1,\infty] \text{ s.t. } \tfrac{1}{p}+\tfrac{1}{q} = 1.$$
But ...

**3**

votes

**1**answer

118 views

### Is this function of a matrix convex?

Let $\mathcal{N}_{n}$ be the set of symmetric nonnegative irreducible matrices. For a matrix $A \in \mathcal{N}_{n}$ let $v^{A}$ be its Perron vector, normalized so that $||v^{A}||_{2}=1$.
Define the ...

**1**

vote

**0**answers

333 views

### What does this notation mean: matrix norm with a two-number subscript

I recently came across this notation, without explanation, in a paper:
$||\mathbf{W}||_{2,1}$
From the context, I know that $\mathbf{W}$ is a matrix, which could be any size, and that ...

**3**

votes

**1**answer

207 views

### Matrix norms / eigenvalues / singular values / another thing

OK, here is what is probably a stupid question.
Let $M$ be a non-symmetric real matrix: for example, the shear matrix
$\left( \begin{array}{cc} 1 & 1 \\\ 0 & 1 \end{array} \right)$.
There ...

**5**

votes

**1**answer

243 views

### The minimal norm of a shifted stochastic matrix

Hello,
Given a row-stochastic matrix $M$ with singular values $\sigma_{1}\geq\ldots\geq\sigma_{n}$, I am looking for an upper bound on the expression: $\min_{\alpha}\parallel M- ...

**2**

votes

**2**answers

247 views

### On matrix norms

It is standard to define an induced matrix norm $|||\cdot|||$ from a vector norm $||\cdot||$ in this way:
$|||A|||=\max_{x \neq 0}{\frac{||Ax||}{||x||}}$.
Suppose we define a different function of ...

**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 ...

**1**

vote

**0**answers

145 views

### What is the Birkhoff norm of a Perron vector?

Let $A$ be a positive matrix. What is known about the Birkhoff norm of its Perron vector?
By the Birkhoff norm of a vector $x$ I refer to the quantity $\frac{\max{x}}{\min{x}}$.
P.S. This is ...

**3**

votes

**3**answers

731 views

### upper bounds on a certain matrix norm

Is there some simple upper bound on $||(B^{-1}+A^{-1})^{-1}||$, where $A,B$ are $n \times n$ symmetric matrices?

**17**

votes

**1**answer

619 views

### Extremal points of the unit ball of M_n(R) …

The unit ball of ${\bf M}_n(\mathbb R)$ is a compact convex subset. As such, it is (Krein-Milman theorem) the convex envelop of its extremal points. So far, so good; but the unit ball depends of the ...

**8**

votes

**0**answers

450 views

### Bounding sum of first singular values squared for Kronecker sum of traceless matrices

Let $A$ and $B$ be $4\times4$ traceless matrices with Hilbert-Schmidt norms summing up to $1/4$, i.e.
$$\text{Tr}\left[ A\right]=\text{Tr}\left[ B\right] = 0,\qquad\text{Tr}\left[ A^\dagger A + ...

**14**

votes

**4**answers

524 views

### Bounding the commutator [A,B] in terms of the numerical radius

Given a norm $N$ over ${\bf M}_n(\mathbb C)$, it is a natural question to find the best constant $C_N$ such that
$$N([A,B])\le C_N N(A)N(B),\qquad\forall A,B\in{\bf M}_n(\mathbb C).$$
The answer is ...