**16**

votes

**2**answers

1k views

### Optimizing the condition number

Suppose I have a set $S$ of $N$ vectors in $W=\mathbb{R}^m,$ with $N \gg m.$ I want to choose a subset $\{v_1, \dots, v_m\}$ of $S$ in such a way that the condition number of the matrix with columns ...

**7**

votes

**5**answers

353 views

### Packing obtuse vectors in $\mathbb{R}^d$

I came across this attractive theorem:
Theorem. In $\mathbb{R}^d$, there can be at most $d+1$ vectors that
form an obtuse angle with one another.
This was proved1 as a corollary of a lemma about ...

**6**

votes

**2**answers

158 views

### Existence and characterization of transitive matrices?

We call a matrix $M \in \mathbb{R}^{d \times d}$ transitive if it satisfies the following:
For any three vectors $u, v, w$ in $\mathbb{R}^d$. If $u^T M v > 0$ and $v^T M w > 0$ then $u^TMw ...

**4**

votes

**2**answers

762 views

### How to calculate the square root of matrix $A+B$ perturbatively?

$A=diag\{\lambda_1,...,\lambda_n\}$ and $\lambda_i>0$, $B$ is a positive definite symmetric matrix and $max\{B_{ij} \}\ll min\{\lambda_i\}$
Note that the perturbative calculation of square root ...

**30**

votes

**3**answers

3k views

### A curious determinantal inequality

In my study, I come across the following curious inequality, which I do not know a proof yet (so I am asking it here).
Let $A, B$ be $n\times n$ (Hermitian) positive definite matrices. It is very ...

**14**

votes

**1**answer

499 views

### Operator norms of circulant matrices

The definition and basic properties of circulant matrices can be found here: http://en.wikipedia.org/wiki/Circulant_matrix.
For complex numbers $a_1,\ldots,a_n$, I will use the notation
$$
...

**4**

votes

**4**answers

1k views

### The multiplicity of the max eigenvalue in matrix multiplication

Suppose that eigenvalues of two real square matrix $A$ and $B$ are $1 = \lambda^A_1 > \lambda^A_2 \geq \ldots \geq \lambda^A_n > 0 $ and $1 = \lambda^B_1 > \lambda^B_2 \geq \ldots \geq ...

**3**

votes

**1**answer

367 views

### Eigenvectors as continuous functions of matrix - diagonal perturbations

The general question has been treated here, and the response was negative. My question is about more particular perturbations. The counterexamples given in the previous question have variations not ...

**7**

votes

**1**answer

620 views

### The singular values of the Hilbert matrix

The $n\times n$ Hilbert matrix $H$ is defined as
$H_{ij} = \frac{1}{i+j-1}, \qquad 1\leq i,j\leq n$
What is known about the singular values $\sigma_1\geq\ldots\geq \sigma_n$ of $H$?
For example, ...

**6**

votes

**1**answer

415 views

### Is the p-norm of a matrix strictly log-convex?

Let $A$ be a $n\times n$-matrix. We let $\|A\|_p$ denote the norm of $A$ when considered as a linear operator on $\ell^p(\{1,2,\ldots,n\})$, that is,
$$
\|A\|_p = \sup_{x\neq ...

**5**

votes

**3**answers

275 views

### Norm of triangular truncation operator on rank deficient matrices

Let $T_{n\times n}$ be a triangular truncation matrix, i.e.
$$T_{i,j}=\begin{cases}1 & i\ge j\\ 0 & i<j \end{cases}$$
It is known that for arbitrary $A_{n\times n}$
$$\|T\circ ...

**3**

votes

**1**answer

89 views

### Wiener-Hopf factorization of matrices

Given a $2\times2$ matrix, which entries are functions in the complex plane $$\hat{A}(z)=\left(\begin{array}{cc}a(z)&b(z)\\c(z)&d(z)\end{array}\right)$$Where $a(z),b(z),c(z)$ and $d(z)$ are ...

**3**

votes

**2**answers

407 views

### How to solve a non-homogeneous quadratic matrix equation?

I am looking to solve the following matrix equation for $G$
$$GHG + M = 0$$
where $G$, $H$, and $M$ are square, symmetric, real matrices. $H$ is negative-definite and $M$ is positive-definite. $G$ ...

**3**

votes

**1**answer

567 views

### SDP formulation of noisy low rank matrix completion

Exact low rank matrix completion using nuclear norm minimization can be formulated as a semidefinite program (SDP). Following the notation in the paper, a convex problem for noisy matrix completion ...

**1**

vote

**0**answers

96 views

### Boundary of pseudospectra

Suppose:
$B_i \in \mathbb{C}^{n \times n}$, $0<w_i\in \mathbb{R}$ $(i = 0,1,2,\ldots,m)$
${\rm P}(x) ={\rm{B}_m} x ^m + \cdots + B_1 x + B_0$ is a matrix polynomial, and $x $ is a complex ...

**1**

vote

**0**answers

178 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 ...

**0**

votes

**1**answer

148 views

### Relation between the subordinate norm and the spectral radius of a matrix

Let's define the following subordinate norm of a $(NM \times NM)$ matrix A norm as follows
\begin{eqnarray*}
||A||_{2,b} = \mathrm{max}_{x \in \mathbb{C}^{NM}} \left \{ \frac{||A x||_b}{||x||_2} ...