**-4**

votes

**0**answers

24 views

### how to select a submatrix from a matrix [on hold]

suppose I have matrix, M is as such [A,B;C,D]; and I want to expand the M matrix as such,
M=P1*M*P1'+P2*M*P2'+P3*M*P3'+P4*M*P4';
where,
P1*M*P1'=[A,0;0,0];
P2*M*P2'=[0,B;0,0];
then other two.
It ...

**2**

votes

**2**answers

495 views

### Multinomial transformation for matrices

Suppose we have a vector of probabilities $\mathbf{p}=(p_1,...,p_n)$, where $p_i>0$ for $i=1,...n$ and $\sum p_i=1$. Define new vector $\mathbf{r}=(r_1,...,r_{n-1})$ in a following way:
...

**0**

votes

**0**answers

61 views

### Advanced use of commutation matrices [on hold]

I am aware of matrix operators vec and kronecker product, commutation matrices and various related identities like stated in ...

**12**

votes

**2**answers

238 views

### Factorization of a matrix as a product of a symmetric and a skew-symmetric matrix

When can an $n\times n$ matrix $M$ be written as a product $M=AB$, where $A^T=A$ and $B^T=-B$?
For example, a necessary condition is that the trace of $M$ vanishes. In this case, it is easy ...

**2**

votes

**0**answers

59 views

### When is $\left[\begin{smallmatrix} D_1 & B \\\\ -B^T & D_2 \end{smallmatrix} \right]$ $\mathbb{R}$-diagonalizable?

Is there some block-wise characterization of $\mathbb{R}$-diagonalizability (by similarities) of
$$\begin{bmatrix} D_1 & B \\\\ -B^T & D_2 \end{bmatrix},$$
where $D_1$ and $D_2$ are real ...

**0**

votes

**0**answers

14 views

### Reducing the degrees of freedom in unitary columns

Let $U = diag([U_1, U_2, ..., U_N])$ be a block-diagonal $NM \times NM$ unitary matrix, where each $U_j$ is a unitary $M \times M$ matrix.
Furthermore, let $Q_e = kron(I_M, Q)$ be the Kronecker ...

**7**

votes

**2**answers

154 views

### Computer Algebra Systems that support variable sized matrices

I'm familiar with sympy, the matlab symbolic package, reduce, and have tried out a few other computer algebra systems. However, as far as I can tell, none of them seem to be able to do algebra on ...

**4**

votes

**2**answers

246 views

### unitary reduction of $q$-normal matrices

The unitary reduction of normal matrices is a well-known fact: if $A\in M_n(\mathbb C)$ commutes with its Hermitian adjoint $A^*$, then there exists a unitary $U\in\mathbb U_n$ and a diagonal matrix ...

**-2**

votes

**0**answers

25 views

### $M = P_1U_1$ and $N = P_2U_2$ ,$M $ and $N$ are unitarily equivalent $ \Rightarrow $ $P_1$ and $P_2$ are unitarily similar [closed]

Let $M, N ∈ M_n$. Let $M = P_1U_1$ and $N = P_2U_2$ be polar decompositions. ($P_i$ is positive semidefinite and $U_i$ is unitary matrix).
Suppose $M $ and $N$ are unitarily equivalent.
Why are ...

**3**

votes

**1**answer

101 views

### The action of $GL(\mathbb{R}^{n})\otimes GL(\mathbb{R}^{m})$ on $\mathbb{R}P^{(mn-1)}$

Motivated by the following RG question we ask a related question as follows:
We identify $\mathbb{R}^{n} \otimes \mathbb{R}^{m}$ with $\mathbb{R}^{mn}$. We define $GL(\mathbb{R}^{n})\otimes ...

**7**

votes

**0**answers

63 views

### Which ordering of factors is needed to obtain this kind of determinantal inequalities?

Let $A$ and $B$ be $n\times n$ Hermitian positive definite matrices. The curious determinantal inequality given here, which can be stated as $$\det (A^{4}+ ABBA+BAAB+B^{4})\ge\det(A^{4}+ ...

**0**

votes

**0**answers

35 views

### Expected value of minimum rank of random matrices

I have $n$ random vectors ${\bf r}_i$ for $i=1,2,\dots,n$ each with dimension $1 \times m$ and $n$ random matrices ${\bf S}_i$ for $i=1,2,\dots,n$ each with dimension $M \times m$. The elements of ...

**8**

votes

**0**answers

189 views

### Standard polynomials applied to matrices (bis)

The standard polynomial in $r$ non-commuting indeterminates $x_1,\ldots,x_r$ is defined by
$${\mathcal S}_r(x_1,\ldots,x_r):=\sum_{\sigma\in S_r}\epsilon(\sigma)x_{\sigma(1)}x_{\sigma(2)}\cdots ...

**0**

votes

**2**answers

120 views

### The condition number of a scaled Vandermonde matrix

Let $V(x_1,..,x_n)$ be the Vandermonde matrix induced by $x_1,..,x_n$, and
let $\tilde{V} := V(\frac{x_1}{h},...,\frac{x_n}{h})$.
My intuition says that the condition number should be invariant under ...

**8**

votes

**2**answers

152 views

### generalizations of Vandermonde matrix to high dimensions

Let $x_1,x_2,\cdots,x_n\in\mathbb{R} $ or $\mathbb{C}$. By the non-degeneracy of Vandermonde matrix
the maps
$$
f: \mathbb{R}\longrightarrow\mathbb{R}^n,$$ $$
x\longmapsto ...

**9**

votes

**1**answer

193 views

### The intuition behind the Hilbert projective metric and the Perron Frobenius Theorem

Recently I have read a proof of the Perron Frobenius Theorem for positive aperiodic matrices. In this proof, the trick is to put a metric in the "positive quadrant" of $\mathbb{R}^n$, ...

**9**

votes

**3**answers

311 views

### Integer matrix that does not belong to a free group of rank 2

I'm given two matrices in $SL_2(\mathbb{Z})$
$$
A = \left(\begin{array}{cc}
2 & 3\\
3 & 5
\end{array}\right), \ \
B = \left(\begin{array}{cc}
5 & 3\\
3 ...

**6**

votes

**1**answer

139 views

### Are the integer matrices in SO(3,2) “boundedly generated”?

Let $G$ be the subgroup of integer matrices in $\mathrm{SO}(3,2)$.
(The invertible linear maps from a $5$ dimensional real vector space to itself which leave invariant a nondegenerate symmetric ...

**1**

vote

**0**answers

24 views

### Approximating logarithm of operator norm of a submatrix

This is a matrix approximation-theoretic question that arose in my study of Markov chains.
Let $\|B\|:=\sup_{v: \|v\|=1} \|Bv\|$ denote the operator norm of a matrix $B.$
Let $\sqrt{B}$ be the ...

**1**

vote

**0**answers

31 views

### Efficient SVD of a matrix without some of the columns

I have a matrix $A \in \mathbb{R}^{p \times q}$ of rank $r$ and its SVD decomposition, i.e,
$$
A = U S V^\top,
$$
where $U \in \mathbb{R}^{p \times r}$ and $V \in \mathbb{R}^{q \times r}$ are ...

**17**

votes

**1**answer

375 views

### Almost commuting unitary matrices

Suppose that $A_1,\dots, A_k$ are unitary matrices such that any two of them can be approximated by commuting unitary matrices. i.e. for any $i$ and $j$, there are unitary matrices $A_i'$ and $A_j'$ ...

**3**

votes

**1**answer

228 views

### Bounded domain matrix factorization

Assume we're trying to find $A\in [-1,1]^{n\times d}, B\in [-1,1]^{m\times d}$, from an observed matrix $C\in [-d,d]^{n\times m}$, where $C=AB^T$.
The goal is to return $\widehat A, \widehat B$ such ...

**1**

vote

**0**answers

95 views

### Lower bound on the value $\textbf{1}^Tx$ such as $Ax\geq b$

The problem may be formulated as follows:
We are given a set of $m$ positive numbers $\{b_1,...,b_m\}$ and a set of $n$ positive numbers $\{v_1,...,v_n\}$. We have $v_j\leq K$, $j=1,...,n$, for a ...

**-1**

votes

**1**answer

32 views

### finding a unitary submatrix inside a random matrix

Let $\mathbf{R} \in \mathbb{C}^{~m \times n} $ with $m \leq n $ be a random matrix, whose entries are i.i.d zero mean random variables with circularly symmetric Normal distribution. Let where $r$ be ...

**2**

votes

**0**answers

25 views

### Nonconvex function on the singular value vectors of projected matrix

For a nonconvex function $p_\lambda(\cdot)$, with hyper-parameter $\lambda$, it can be decomposed into two parts that $p_\lambda(t) = \lambda |t| + q_\lambda(t)$, where $q_\lambda(t)$ is the concave ...

**12**

votes

**1**answer

334 views

### Hlawka inequality for determinants of positive definite matrices

It is mentioned here that if $A, B, C\in M_{n}(\mathbb C)$ are positive semidefinite, then $$\det (A+B+C)+\det C\ge \det (A+C)+\det (B+C)$$ (quoted from this article) and the special case ($C=\bf 0$) ...

**-1**

votes

**1**answer

169 views

### How bad could $\|A^k\|$ be when $\rho(A) < 1-\delta$ [closed]

(Sorry, I do hate editing this many many times but let me try the last time)
Gelfand's formula says that
$$\lim_{k\rightarrow \infty} \|A^k\|^{1/k} = \rho(A)$$
I am wondering whether there is any ...

**0**

votes

**0**answers

24 views

### transform a matrix into block circulant form

What's known about transforming a matrix into block circulant form?
Given an $K \times N$ matrix $M$ (over GF(2) for now although the case
over ${\mathbb C}$ or ${\mathbb R}$ are also of interest), ...

**4**

votes

**1**answer

197 views

### Hankel matrix commuting with a Jacobi matrix

Assume the semi-infinite Hankel matrix $H$ with entries
$$H_{i,j}=\alpha_{i+j-1}, \quad i,j\geq1,$$
where $\alpha_{n}\in\mathbb{R}$, is given. It turns out that in some special situations a ...

**3**

votes

**1**answer

177 views

### Prediction with positive weights?

Consider a covariance function (positive definite function) on $\mathbb{Z}$:
$$
\gamma(k)=(1+|k|)^{-\alpha},\quad \alpha>0.
$$
It is guaranteed to be positive definite by Polya's criterion ...

**1**

vote

**0**answers

124 views

### Is there a brute force method for determining irreducible representations?

Suppose I have some groups $G_1$, $G_2$, $G_3$, etc... Then the direct product is given by $G = G_1 \times G_2 \times G_3 \ldots$
I know that the sub-representations of a reducible representation ...

**6**

votes

**1**answer

139 views

### Exchange determinant and integral of a matrix-valued function

Assume $A(x)=(a_{ij}(x))_{k\times k}$ is a Hermitian matrix function on some manifold $M$, is there any inequality relates the integral of its determinant $\int_M det(A)$ and the determinant of its ...

**11**

votes

**1**answer

218 views

### Have “sturdy squares” been studied before?

Over on PPCG I've just made a question that involves arranging the numbers 1 to 9 on a 3 by 3 grid such that every 2 by 2 subgrid has the same sum. I'm calling such 3 by 3 grids and their N by N ...

**148**

votes

**21**answers

21k views

### Geometric Interpretation of Trace

This afternoon I was speaking with some graduate students in the department and we came to the following quandry;
Is there a geometric interpretation of the trace of a matrix?
This question ...

**9**

votes

**2**answers

244 views

### Kernel of skew-symmetric matrix of rank $n-1$ with $n$ odd: is this a known result?

When $n$ is odd, the kernel of a skew-symmetric matrix $M$ of size $n\times n$ and rank $n-1$ is the span of $v$, where $v$ is a vector whose $i$-th component is the Pfaffian of the matrix obtained by ...

**5**

votes

**1**answer

95 views

### What is the criterion for a matrix containing vectors and their permutations being invertible?

Consider the matrix $A\in\mathbb{R}^{m\times 2m}$. Let any arbitrary choice of $m$ columns of $A$ be linearly independent. Together with a permutation $P\in\mathcal{P_{2m}}$, one can build the matrix ...

**7**

votes

**2**answers

615 views

### When is the determinant a Morse function?

This might be ridiculously obvious, but...
For each $n \in \mathbb{N}$, let $M_n$ denote the manifold of $n \times n$ matrices with real entries. It is well known that the $n$-dimensional determinant ...

**18**

votes

**4**answers

2k views

### Linear transformation that preserves the determinant

It seems "common knowledge" that the following holds:
Let $T$ be a linear transformation on nxn matrices with complex coefficients that preserves the determinant. Then there exists matrices U and V ...

**7**

votes

**0**answers

104 views

### A $k \times n$ matrix with a lot of invertible $k \times k$ submatrices over $\mathbb{F}_2$

In the appendix of the paper by Tolhuizen (
http://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=841182) there is a very fast and easy probabilistic proof that for $k=cn$ where $c \in [0,1]$ is fixed, ...

**3**

votes

**0**answers

74 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 ...

**1**

vote

**0**answers

78 views

### For of a special case of $Ax>=b$, are there always integer solutions?

The input consists of two integers $n\geq 2$ and $K\geq 2$, and a vector of positive integers $b$ of size $n$. We assume that $\sum_i b_i$ is a multiple of $K$.
The output is a matrix $A(n\times n)$ ...

**0**

votes

**0**answers

83 views

### Proof sought that a certain function is approximatively 3-dimensional

Let $D_n(x)$ be a diagonal matrix of size $N\times N$ where the $k$th element is $\exp(2\pi\jmath x(n+(k-1)/N)$.
Let $P_n$ be a random diagonal $N\times N$ matrix where each diagonal element is a ...

**1**

vote

**0**answers

39 views

### Product of elementary divisors

Let $A$ be an $(m \times n)$ integer matrix (if it helps, we can assume that a is a square matrix). Let $d_i,\ldots,d_s$ be the elementary divisors of $A$. I am interested in the product ...

**7**

votes

**0**answers

229 views

### Product $PVPVP$ is elementwise nonnegative?

Let $P\in \mathbb{R}^{n\times n}$ be the inverse of a positive definite M-matrix, i.e. the off-diagonal entries of $P$ are non-positive, and $V\in \mathbb{R}^{n\times n}$ be any diagonal matrix. Prove ...

**0**

votes

**0**answers

35 views

### log convexity for the norm of vector-valued function

Log convexity of various functions defined on the space of Hermitian matrices plays an important role in matrix analysis and probability theory.
Given $v \in \mathbb{C}^n$, $D$ a diagonal matrix with ...

**9**

votes

**1**answer

340 views

### A maximal element, where Schur gives a minimal element

Let me recall a result due to I. Schur, which I learnt from F. Goldberg's answer to my MO question Hadamard-like inequalites for positive definite symmetric matrices. If $H$ is a subgroup of $\frak ...

**1**

vote

**0**answers

63 views

### On triangular Toeplitz matrices

Let $R(x)$ be the upper triangular Toeplitz matrix with first row $x$, so that $R_{ik}=x_{k-i+1}$ if $i\le k$ and $R_{ik}=0$ otherwise. Let $N(n)$ be the smallest number $N$ such that there exist ...

**4**

votes

**0**answers

60 views

### How does scaling rows to sum to 1, of a positive matrix change the perron vector?

Reposting from math.sx due to lack of response. Let $A$ be a $N\times N$ positive matrix such that $A_{ij}>0$. By Perron-Frobenius theorem, there is a unique positive left eigenvector called Perron ...

**3**

votes

**0**answers

87 views

### Relating Numerical Range and Perron-frobenius theorem for positive matrices?

Let $A$ be any matrix with all entries positive (which means Perron-Frobenius theorem can be applied). Then its numerical range is defined as the set of complex numbers
$$W(A)=\{x^HAx\lvert ...

**5**

votes

**0**answers

322 views

### How the idea of adjugate matrix has been designed? [closed]

I can understand the adjugate matrix and the motivation of that to find the inverse, but I can't see how this idea was invented by mathematicians. It's just brilliance or someone understand how the ...