**2**

votes

**1**answer

163 views

### Extracting a full rank matrix from a 0-1 matrix

If $A$ is a $n\!\times\!n$ $0$-$1$ matrix of rank $k<n$. If ever possible, what would be an efficient way of extracting a full rank $k\!\times\!k$ sub-matrix of $A$ by removing columns and rows of ...

**3**

votes

**1**answer

153 views

### Existence of $SO(n)$-isotropic inner products which are not $O(n)$-isotropic

$\newcommand{\al}{\alpha}$
Let $M_n$ be the space of $n \times n$ real matrices.
Question:
For which $n$, is there an inner product on $M_n$ which satisfies:
$$(*) \, \, \langle Q^TXQ,Q^TYQ ...

**3**

votes

**2**answers

189 views

### functions with orthogonal Jacobian

I'm working on a model that would require to use vectorial functions of $\mathbb{R}^n \rightarrow \mathbb{R}^n$, such that $\forall x, y \in \mathbb{R}^n$, $\lVert \frac{df(x)}{dx}(y) \lVert_2 = ...

**4**

votes

**1**answer

82 views

### Kolmogorov complexity for matrices

In applications one often encounters very large matrices that barely fit in computer memory, if at all. Naturally one wishes to represent those matrices as compactly as possible. Sometimes one even ...

**4**

votes

**0**answers

113 views

### Expectation of a specific random variable on the probability space of $n\times n$ matrices over $\{0,1\}$

Let $\mathcal{G}_{n,\frac{1}{2}}$ be the probability space of $n\times n$ matrices over $\{0,1\}$ and each entry of the matrix is independently equal to 1 with probability $\frac{1}{2}$ and equal to 0 ...

**0**

votes

**1**answer

41 views

### Cross section point of two conics curves

We have $A_i , B_i , C_i , D_i , E_i ,F_i, \ (i=1, 2) $.
We want to find $ (u,v) \in \mathbb{R}^2$ satisfying
\begin{equation}
A_1 u^2 + B_1 uv + C_1 v^2 + D_1 u + E_1 v +F_1 =0 \\
A_2 u^2 + B_2 ...

**2**

votes

**1**answer

149 views

### the pfaffian-adjugate and its counterparts for matrices odd size

Let $R$ be a commutative ring (zero characteristic). Take a skew-symmetric matrix $A\in Mat^{skew-sym}(n,R)$.
If $n$ is even, then $\det(A)=Pf^2(A)$ and there exists the "Pfaffian adjugate/adjoint" ...

**11**

votes

**1**answer

441 views

### An inequality for the spectral radius of matrices used by J. Bochi

I am interested in the history of an inequality for the spectral radius of a $d\times d$ real or complex matrix, which occurs in Jairo Bochi's 2002 article Inequalities for numerical invariants of ...

**0**

votes

**0**answers

82 views

### Primitivity of $AA^\top$

Let $A\in\mathbb{R}^{n\times n}$ be a non-negative and irreducible matrix. Consider $B:=AA^\top$. It can be proved (I can post a proof if needed) that the following condition is necessary and ...

**0**

votes

**0**answers

31 views

### A class of unimodular parametrization

Is there a parametrization of set of matrices $\mathcal M\subseteq\Bbb Z[x_1,\dots,x_{m}]^{n\times n}$ such that $\forall f:\{-1,+1\}^{m}\rightarrow\{-1,+1\}$ $\exists M\in\mathcal M$ such that ...

**0**

votes

**0**answers

33 views

### If $A$ is doubly stochastic and reducible. Why is $A$ permutation similar to a matrix of the form $A_1 ⊕ A_2$

If $A \in M_n$ is doubly stochastic and reducible.
Why is $A$ permutation similar
to a matrix of the form $A_1 ⊕ A_2$, in which both $A_1$ and $A_2$ are doubly stochastic?

**1**

vote

**0**answers

75 views

### Let $M = \frac{1}{2}(A + {A^T})$ be real symmetric nonnegative matrix. Why does $\rho (A) \le {\lambda _{\max }}(M)$?

Let $A \in M_n$ be nonnegative, and consider the real symmetric nonnegative matrix
$M = \frac{1}{2}(A + {A^T})$.
Why does $\rho (A) \le {\lambda _{\max }}(M)$?

**-1**

votes

**1**answer

175 views

### FInd smallest value $r$ such that a $n\times r$ matrix exists [closed]

The input of my problem is an integer $n\geq 3$.
The output is an integer $r\geq 1$ which must be as small as possible such that there is a $(n\times r)$ matrix verifying the following constraints:
...

**1**

vote

**0**answers

18 views

### How can I filter the effects of a variable from a correlation matrix?

I have a correlation matrix (it contains 500 columns and 500 rows) and I would like to make an other correlation matrix in which one variable (and its influences) is filtered from the initial matrix. ...

**1**

vote

**1**answer

79 views

### Multiplicatively closed subsets of $\mathbb{C}^{n \times n}$

While dealing with another problem I saw that I need to classify subspaces $V$ of $\mathbb{C}^{n \times n}$ that are multiplicatively closed.
So to get an idea of the nature of the subspaces I ...

**0**

votes

**1**answer

116 views

### Standard rational functions from matrices

In linear algebra we get introduced to standard polynomials that are associated to matrices such as characteristic polynomials and determinants.
What are some of the standard rational functions that ...

**0**

votes

**1**answer

106 views

### Number of turning points on a nondecreasing $n^2 \times n^2$ matrix

Given a integer value $n$, we generate a $n^2 \times n^2$ integer matrix $M$ in the following way.
Each ceil has value range $[1~n]$
In each row, the value is nondecreasing. E.g. $M[i, j] \leq M[i, ...

**1**

vote

**0**answers

58 views

### Decomposing a matrix into the tensor product of a permutation and orthogonal matrix

Suppose I have a square matrix $A \in \mathbb{R}^{mn \times mn}$. I want to find
$$\arg \min_{P, Q} \|A - P \otimes Q\|_F$$
where $P$ is an $m \times m$ permutation matrix and $Q$ is an $n \times ...

**7**

votes

**2**answers

295 views

### Covering the zeros of 0/1 matrix with submatrices

The matrices I am dealing with are $n\times n$ of the following type (with $n=7$):
$M_7=\begin{pmatrix}1&0&0&0&0&0&1 \\ 1&1&0&0&0&0&0 \\ ...

**3**

votes

**1**answer

171 views

### On a determinantal equality

In my study, I come across the following curious equality, which I do not know a proof yet (so I am asking it here).
Let $k$, $l\in \Bbb Z$ be fixed, $m$ --- the size of the below matrix $M$ --- is ...

**1**

vote

**0**answers

72 views

### A question on Perron–Frobenius theorem [closed]

Let $A \in M_n$ is nonnegative(all $a_{ij}\ge0$).
Suppose $A$ has a nonnegative eigenvector(all entries$\ge0$ ) with $r ≥ 1$ positive entries and $n − r$ zero entries.
Why is there a permutation ...

**4**

votes

**1**answer

71 views

### Robust generalization of matrix rank

I am looking for robust generalizations of matrix rank.
Think of the the following problem: A big matrix of low rank is perturbed by random noise, such that it becomes a full-rank matrix. Is there a ...

**2**

votes

**1**answer

97 views

### Heuristics for counting degrees of freedom

I have recently learned about the representation theorem for isotropic,
linear operators, which says the following:
Defintion:
Let $M_n$ be the vector space of $n \times n$ real matrices. We say a ...

**4**

votes

**1**answer

139 views

### What is the best algorithm for even rank magic square?

Magic square is a $n*n$ matrix with numbers of $1,2,...,n^2$ and has the property that sum of any row and any column and sum of main diameter and
adjunct diameter is identical. There exists a very ...

**5**

votes

**2**answers

380 views

### Does the antidiagonal in this square matrix always contain a prime?

Does the antidiagonal in the square matrix with entries $1,2,\ldots,n^2$ row by row in that order always contain a prime?
For example:
For n=2: $\begin{bmatrix}1 & 2\\3 & 4\end{bmatrix}$ ...

**3**

votes

**1**answer

88 views

### Similarity transformation of transition matrix of reversible Markov chain (reference request)

If $P$ is the transition matrix of a reversible Markov chain, and $\pi$ is its stationary distribution, and let $R$ be defined by:
$$R_{ij} = \sqrt{\frac{\pi_i}{\pi_j}}P_{ij}~.$$
By reversibility, ...

**0**

votes

**0**answers

247 views

### When does this matrix have full rank?

Suppose $\mathbf{B}\in\left[0,1\right]^{T\times M}$ is a binary matrix,
$\mathbf{B}_{i}$ is a column of $\mathbf{B}$, and $\mathbf{X}\in\mathbb{R}^{N\times T}$
is a matrix where the columns are ...

**0**

votes

**2**answers

83 views

### Simultaneous special orthogonal similarity problem

Given matrices $A,B,C,D\in\Bbb K^{n\times n}$ where $\Bbb K$ is a ring is there an efficient technique to compute set $O$ with $OO'=I$ where $'$ is transpose and $\mathsf{Det}(O)=\pm1$ such that
...

**0**

votes

**1**answer

38 views

### Making a real matrix positive definite by replacing nonzero and nondiagonal entries with arbitrary nonzero reals

Let $A$ be real square matrix.
Let $\mathcal{F}(A)$ be the set of real matrices $A'$ of the same size such that $A'_{ii}=A_{ii}$ for all $i$, and for all $i,j$, $A_{ij}=0\Rightarrow A'_{ij}=0 \land ...

**5**

votes

**1**answer

208 views

### A partition of the set of all $n\times n\ (0,1)$-matrices

Let $S_n$ be the set of all the $n\times n\ (0,1)$-matrices and divide $S_n$ into two sets as follows:
$A_n=\{M\in S_n:$ there exist a row and a column of $M$ such that the sum of the row is equal to ...

**0**

votes

**0**answers

97 views

### A linear combination problem

Given $0/1$ $n\times n$ matrix $M$.
Suppose we have $2$ vectors $\lambda,\mu\in\Bbb R^{1\times n}$ such that both
$$\lambda M\in\{0,1\}^{1\times n}$$
$$M\mu'\in\{0,1\}^{n\times 1}$$
holds with $'$ ...

**5**

votes

**0**answers

103 views

### When is a Hermitian matrix of the form $g^*g$ for some matrix $g$

I tried asking this question on Math StackExchange and didn't get any replies. I was hoping that maybe someone here could help, sorry for duplicating.
I'm trying to figure out some properties of ...

**2**

votes

**1**answer

88 views

### Eigenvalues of product of symplectic matrices

I have 2 symplectic matrices $X_{1},X_{2} \in \mathbb{R}^{2n\times2n}$. The matrix $X=X_{1} \cdot X_{2}$ is also symplectic.
Question: Are there any theorems which allow me to express eigenvalues of ...

**1**

vote

**0**answers

32 views

### Most accurate separation of tensor product of matrices

Given a matrix $A \in \mathbb{R}^{n^m \times n^m}$, how I can find a set of $m$ matrices $\{B_i\}$ such that
$$\arg \min_{\{B_i\, \in\, \mathbb{R}^{n \times n}\}} \|A - B_1 \otimes \ldots \otimes ...

**3**

votes

**2**answers

271 views

### Is it always possible to “separate” the eigenvalues of an integer matrix?

Call a square matrix Galois-irreducible if all its eigenvalues are Galois conjugates of each other.
Let $M$ be an integer $n\times n$ matrix which is not Galois-irreducible. Is it always ...

**4**

votes

**3**answers

157 views

### Fast Upper Triangular Matrix Exponentiation

Let $Q_n$ be a $n\times n$ matrix with $Q_n=\begin{pmatrix} -\lambda_1-\mu_1 & \lambda_1 & 0 & \cdots\\ 0 & -\lambda_2-\mu_2 & \lambda_2 & \cdots\\ \vdots & \vdots & ...

**0**

votes

**1**answer

51 views

### Approximate $\mathbf{G}=(a\mathbf{H}+\mathbf{M})^+$ by Taylor expansion [closed]

Suppose we have a complex matrix $\mathbf{M}$. Let $\mathbf{M}^+=(\mathbf{M}^*\mathbf{M})^{-1}\mathbf{M}^*$ be the pseudo-inverse of $\mathbf{M}$, where $^*$ denotes the conjugate transpose. Let ...

**0**

votes

**1**answer

82 views

### A particular Lie algebra $L_{n}$ and (various) lie groups whose Lie algebra is isomorphic to $L_{n}$

We define $$L_{n}=\{A=(a_{ij})\in M_{n}(\mathbb{R})\mid \sum_{i=1}^{n} a_{ij}=0 \;\;\;\text{for every fixed j}\}$$
This is a Lie subalgebra of $M_{n}(\mathbb{R})$.
A dynamic-geometric proof for ...

**6**

votes

**1**answer

66 views

### Least-squares solution of systems of Sylvester equations

The Sylvester equation $AX+XB=C$ has been studied quite a lot and there are known algorithms for solving it.
But has the situation where (an over-determined) system of equations $A_{i}X+XB_{i}=C_{i}$ ...

**1**

vote

**1**answer

181 views

### On Knot Equivalence problem statement

How is the knot equivalence problem represented?
By this I mean I am looking for an analogy that compares with graph equivalence. For graph equivalence, we have two graphs $G_1$ and $G_2$ with ...

**4**

votes

**0**answers

168 views

### Existence or construction of a sequence of orthogonal matrices with three properties

This is a problem that I encountered during my research, and I have spent a good amount of time on it without success. So I am reaching out for help ....
Any pointers or suggestions are appreicated!
...

**5**

votes

**1**answer

152 views

### Sum of the absolute eigenvalues of A>=B

Kindly help me to prove/disprove the following statement.
Let $A$ be a symmetric matrix of order $n \times n$ with all the diagonal entry equal to $0$, and other non-diagonal entry equal to $k$ ...

**3**

votes

**3**answers

142 views

### Can a block matrix with at least 3 zero blocks of different size on the diagonal and 1's everywhere else have only integer eigenvalues?

Let $M=\begin{pmatrix}
\begin{array}{cccccccc}
0 & 0 & 1 & 1 & 1 & 1 & 1 &1\\
0 & 0 & 1 & 1 & 1 & 1 & 1 &1\\
1 & 1 & 0 & 0 & ...

**4**

votes

**2**answers

159 views

### Non-asympototic version of Gelfand's formula

Let $A$ be a $n\times n$ matrix. Let $\|A\|$ be the spectral norm of $A$, and $\rho(A)$ be the spectral radius. I am wondering whether the following statement is true.
There exists universal ...

**6**

votes

**2**answers

394 views

### Parametrization of positive semidefinite matrices

We know that a real, symmetric, positive definite matrix $A$ of size $n\times n$ can be parametrized by a vector $\theta$ of $\frac{n(n+1)}{2}$ parameters thanks to the Cholesky decomposition:
$$
A = ...

**0**

votes

**0**answers

81 views

### Is there a metric defined on the product space of orthogonal groups?

If one considers just the orthogonal group, then there is a natural metric given by [1]:
$$\begin{align}
\theta & =\frac{1}{2} \| \log(R_1^{-1}R_2) \| \\
& = \frac{1}{2} \| ...

**-5**

votes

**1**answer

89 views

### Simple bimodule over matrix ring [closed]

Let given not trivial simple $R$- $R$ bimodule $M$, where $R$ - $n\times n$ matrix algebra over field $\mathbf{F}$. Is it true that $M$ is uniquely defined?

**3**

votes

**1**answer

114 views

### Uniqueness of the reduced rank QR decomposition

Let $A$ be a $n\times n$ matrix that is a real, symmetric, positive semidefinite and has rank $r$.
I read about the rank reduced $QR$ decomposition (here for example) as the QR variant where $A=QR$ ...

**12**

votes

**2**answers

335 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 ...

**3**

votes

**0**answers

72 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 ...