**1**

vote

**0**answers

104 views

### An (open?) problem about a sequence of nested principal sub-matrices and their determinants [on hold]

Problem: Let $A$ be a $n \times n$ integer matrix, $\det(A) = \pm 1$. Under which conditions there exist a nested sequence of principal submatrices of size $n$ such that they all have determinant $\pm ...

**4**

votes

**0**answers

64 views

### Geometric interpretation of the Desnanot-Jacobi Identity

Given a square $n\times n$ matrix $M$, let $M_i^j$ denote the $(n-1)\times(n-1)$ matrix obtained from M by omitting the i-th row and j-th column of $M$.
The Desnanot-Jacobi Identity states
...

**-2**

votes

**0**answers

42 views

### What kind of matrix is similar to an irreducible matrix? [on hold]

For example, we know that the set of positive definite matrix is similar to an irreducible matrix. Therefore, the set of such matrices should include the positive definite matrix cone. What kind of ...

**2**

votes

**1**answer

172 views

### Lebesgue measure of set of $y\in\mathbb{R}^n$ such that $x,y,Ay$ are linearly dependent

I've asked this question here on math.stackexchange, but I have been unable to solve this yet, so I'm hoping I can get some advice here.
Consider a vector $x\in \mathbb{R}^n$ and a real $n\times n$ ...

**2**

votes

**1**answer

55 views

### Does Schatten-p (quasi-)norm satisfy the norm inequality for 0<p<1?

I'm reading the paper by ANGELIKA ROHDE AND ALEXANDRE B. TSYBAKOV, ESTIMATION OF HIGH-DIMENSIONAL LOW-RANK MATRICES.
And in the paper, they provide an inequation of the Schatten-p (quasi-)norm, ...

**7**

votes

**0**answers

60 views

### What is the symmetry group fixing norms of elements of a unitary matrix?

Let $N\geq1$ be an integer and let us say that two matrices $U,V\in U(N)$ are related if $|U_{ij}|=|V_{ij}|$ for all indices $1\leq i,j\leq N$.
When exactly are two unitary matrices related in this ...

**-4**

votes

**0**answers

40 views

### Linear Algebra, Finding All Basis for B^n , etc [closed]

Let B^n be the vector space of column vectors with bit entries 0, 1. Write down every basis for B^1 and B^2. How many bases are there for B^3? B^4? Can you make a conjecture for the number of bases ...

**-4**

votes

**0**answers

59 views

### Elementary book on Matrices [migrated]

can anyone recommend a book that focusses on matrix theory at an elementary level? I was never taught matrices in high school (25 years ago) and I'm teaching myself algebra using The Everything Guide ...

**0**

votes

**2**answers

60 views

### Symmetric matrix from a nonsymmetricc matrix

Basically this is a part of a long algorithm to calculate some matrix properties.
Given an upper triangular square matrix R, how can I find an orthonormal matrix W (possibly iteratively) such that WR ...

**1**

vote

**2**answers

116 views

### Norm of a matrix operator with a special structure

Let $\{\alpha_{n}\}_{n\in\mathbb{N}}$ be positive sequence such that
$$\sum_{n=1}^{\infty}\alpha_n<\infty.$$
Question: Is there any chance to evaluate the operator norm of the matrix operator
...

**1**

vote

**1**answer

156 views

### Homotopy type of certain maps on complex grassmanian

$G(k,n)$ is the complex grassmanian which is homeomorphic to the space of projections in $M_{n}(\mathbb{C})$ with trace $k$. So we can Identify $G(k,n)$ with $$\{A\in M_{n}(\mathbb{C})\mid ...

**1**

vote

**1**answer

67 views

### On the least singular values

Let $A$ be a square matrix of size $n \times n$ ($n>2$) and let $B$ be $A$ if we delete the last row and column (size $(n-1) \times (n-1)$). Let $\sigma (A)$ be the least singular value of $A$ and ...

**2**

votes

**4**answers

259 views

### Finding commuting matrices

Is there a procedure for finding all matrices which commute with two given square and complex matrices?
For example, given two elements $A,B \in$ $\mathfrak{su}(4)$ is it possible to find all ...

**2**

votes

**1**answer

69 views

### Equivalence of entrywise 1-norm and Schatten-1 norm

Let $A \in \mathbb{R}^{m\times n}$ and $\|A\| = \sum_{i, j} |A_{i,j}|$.
I am looking for constants $\alpha, \beta \in \mathbb{R}$ such that
$\alpha \|A\| \leq \|A\|_* \leq \beta \|A\|$
The function ...

**1**

vote

**2**answers

121 views

### A certain matrix associated to graphs

I am not very familiar with graph theory, but I need some results for my work. Thus, the question is, whether the following has already been studied and where I can find it. Let $G=(V,E)$ be an graph ...

**2**

votes

**1**answer

53 views

### Decomposing large symmetric banded sparse matrices

I'm investigating 3D image deblurring and one of the approaches I'm interested in is applying spatial regularisation. To do this I have generated a matrix $A$ which encodes the 6-connectivity of each ...

**0**

votes

**1**answer

102 views

### Minimal dimension of a Lie algebra of matrices, with a restrictive property

Let $\mathfrak{g}$ be a sub-Lie-algebra of $\mathfrak{gl}_n(\mathbb{C})$, the Lie algebra of complex $n\times n$ square matrices.
Let us call $(H)$ the hypothesis: for all $x, y\in\mathbb{C}^n$, ...

**0**

votes

**0**answers

27 views

### Is there a term for “ranked distance” matrices?

In a n by n "ranked distance matrix" each element has a rank $r_{ij}$ between 1 and n that indicates it is the $r_{ij}$th smallest element in column $i$ of a corresponding Euclidean distance matrix. ...

**1**

vote

**1**answer

99 views

### Dense symmetric unitary integer matrix?

Can someone give me a nontrivial example of a symmetric unitary integer matrix? I'm looking for something as dense as possible (i.e., not too many 0's); 5 <= size <= 8 would be ideal.

**10**

votes

**1**answer

158 views

### How can we interpret the eigenvalues and eigenvectors of Euclidean Distance Matrices?

I asked this question in Math Stack Exchange earlier here: http://math.stackexchange.com/questions/1199380/what-is-the-intuition-behind-how-can-we-interpret-the-eigenvalues-and-eigenvec and since I ...

**6**

votes

**5**answers

292 views

### The maximal eigenvalue of a symmetric Toeplitz matrix

Let $0\le x\le 1$ be a real number. Denote by $A_n(x)=(a_{ij})$ the $n$ by $n$ matrix such that $a_{ij}=x^{|i-j|}$ and let $\lambda_n(x)$ be the maximal eigenvalue of $A_n(x)$.
Is there any ...

**0**

votes

**0**answers

79 views

### What is wrong with the argument that zero permanent is polynomial?

This Lecture summarizes some well known facts about $\#P$ completeness of permanent.
Given a CNF formula $\phi$ on $n$ variables, they construct
matrix $A$ such that:
$$perm(A)=4^{3m} \#SAT(\phi)$$
...

**4**

votes

**1**answer

149 views

### Represent matrix immanants using Schur functions

For each irreducible character $\chi^\lambda$ of the symmetric group $S_n$, the immanant of an $n\times n$ square matrix $A$ is defined as
\begin{equation*}
d_\lambda(A) := \sum_{\sigma \in S_n} ...

**7**

votes

**0**answers

264 views

### Does this inequality always hold?

Denote the adjacency matrix of a given undirected graph by $g$. It is an $n$-by-$n$ symmetric Boolean matrix with elements on the diagonal to be zero ($n\geq 3$). Let $g_{12}=g_{21}=g_{13}=g_{31}=1$ ...

**2**

votes

**1**answer

66 views

### Are there non-isomorphic graphs with rationally orthogonal similar adjacency matrices?

Let $A_G,A_H$ be the adjacency matrices of two non-isomorphic graphs.
Let $P$ be orthogonal matrix with rational entries.
Is it possible $A_G = P^{-1}A_H P$?
Paper
gives algorithm for ...

**0**

votes

**0**answers

44 views

### Determinant of a Vandermonde matrix of roots of monic polynomial with integer coefficients [migrated]

Let $p(x)=\sum_{i=1}^n a_ix^i$ with $a_i$ an integer for all $i$ and $a_n=1$ such that $p(x)$ has only real roots, and let $\lambda_1,\ldots,\lambda_n$ be the $n$ roots of this polynomial. Then the ...

**5**

votes

**2**answers

247 views

### When are the adjacency matrices of non-isomorphic graphs similar?

From Wikipedia.
In linear algebra, two n-by-n matrices A and B are called similar if
$$ B = P^{-1} A P$$
for some invertible n-by-n matrix $P$.
If $P$ is a permutation matrix, $A$ and $B$ are ...

**13**

votes

**1**answer

458 views

### Enumeration of $0-1$ matrices with determinant $1$

Has the number $f(n)$ of $n \times n$, $0{-}1$ matrices whose determinant is $+1$
been enumerated?
E.g., for $n{=}2$, there are $f(2)=3$ such matrices:
$$
\left(
\begin{array}{cc}
1 & 0 \\
0 ...

**0**

votes

**0**answers

34 views

### Changes in singular Values of matrix when adding row

I know that if a column is added to a matrix then the matrix largest signular value increases and the smallest singular value decreases. That is:
Given matrix $A \in R^{m \text{x} n}$, $m>n$, and ...

**2**

votes

**2**answers

198 views

### quadratic matrix equation

Find all symmetric matrices $X=X^{T}$ such that
\begin{align}
XDX^{T}=-D \quad (1)
\end{align}
where $D\ne 0$ is a real diagonal matrix.
For example, $X=iI$ satisfies $(1)$. Can you get a ...

**3**

votes

**0**answers

78 views

### Bound on the ratio of top 2 eigenvalues

Let $P$ be a $(n+1) \times (n+1)$ stochastic matrix such that $P_{ij}=\tau$ if $i \neq j$ and $P_{ii} = (1 - n\tau)$ where $0<\tau < \frac{1}{n+1}$. It is clear that the largest eigenvalue of ...

**3**

votes

**2**answers

155 views

### What are interesting heuristics of determining how far given matrix is from a singular one?

The condition number and volume of matrix (defined as absolute value of its determinant) are things which come to mind. Is there more?
I think that over the years numerical folks (who are faced with ...

**5**

votes

**0**answers

85 views

### A weak Perron-Frobenius property for sets of positive matrices

A popular form of the Perron-Frobenius theorem states the following result: if $A$ is a $d \times d$ real matrix all of whose entries are positive, then the spectral radius of $A$ is a simple ...

**0**

votes

**0**answers

52 views

### Is the function below convex?

I have the following function $f(X)=(\sum(gmm^2(AX)-2gmm(AX)gmm(B)))||CX-D||^2$
where gmm is Gussian mixtures defined as $gmm(x)=\sum_{i=1}^{K}\omega_{i}\phi(x|\mu_{i},\Sigma_{i})$, $\omega$ is the ...

**0**

votes

**0**answers

88 views

### Range of a trace preserving completely positive projection

I'm working in $M_n(\mathbb{C})$, the algebra of complex $n\times n$ matrices. I managed to build a completely positive, trace preserving, star preserving, projection $P$. That is
$$\text{Tr}(P(A)) = ...

**2**

votes

**2**answers

124 views

### How to calculate one Cauchy type determinant

As we know, a Cauchy determinant of size n admits the following explicit formula:
$$\det \left(\frac{1}{x _i+y _j}\right) _{1\le i,j \le n}=\frac{\prod _{1\le i < j\le n} (x _j-x _i)(y _j-y ...

**1**

vote

**0**answers

146 views

### Is there a method to simultaneously block-diagonalize a set of group matrices?

Assume that you are explicitly given the representation matrices of a group.
How does one go about finding that common basis which will find the irreducible components of all of them simultaneously?
...

**1**

vote

**0**answers

51 views

### What are good bounds on ratios of subdeterminants?

Let $A$ be a symmetric matrix and $A_i$ be the matrix obtained from $A$ by dropping the $i^{th}$ row and column. Then what are some good bounds on the value of $\frac{det(A_i)}{det(A)}$ ?
Using the ...

**5**

votes

**1**answer

245 views

### When is a linear combination of permutation matrices unitary?

Question:
Let $P_\pi$ denote the matrix representation of permutation $\pi$. Consider a linear combination of all $n \times n$ permutation matrices
$$U := \sum_{\pi \in S_n} c_\pi P_\pi$$
where ...

**3**

votes

**0**answers

70 views

### When is a Hankel matrix invertible?

I would like to have some conditions that render a general Hankel matrix of the form
\begin{pmatrix}a_1 & a_2 & a_3 &\cdots & a_n \\ a_2 & a_3 & a_4 & \cdots & a_{n+1} ...

**1**

vote

**2**answers

107 views

### Matrices congruent to each other via a permutation

Consider the collection of all integer matrices and partition them via an equivalence relation $A\sim B\Leftrightarrow \exists$ a permutation matrix $P$ such that $B=PAP^T$. Is some canonical form ...

**1**

vote

**2**answers

137 views

### Inverse of a matrix expression

Let
$$X_i = \left(I - P\left(I - t_it_i^T\right)\right)^{-1}$$
where $P$ is an $N\times N$ matrix and $t_i$ is a vector of $N$ elements.
Is there a way to simplify this expression in order to ...

**0**

votes

**1**answer

115 views

### $\mathbb{Z}_{2}$ -equivariant vector bundles over manifold of rank-$k$ matrices

Edit: I remove the trivial part of the first version, according to comment of Alex Degtyarev
Let $M$ be the manifold of all matrices in $M_{n}(\mathbb{R})$ with fixed rank $0<k<n$. There is ...

**3**

votes

**1**answer

83 views

### Probe permutationally matrix extreme properties

Suppose given $M\in\{0,1\}^{n\times n}$ of rank $r$.
Assume that changing even a single $1$ to $0$ in $M$ raises rank. Does it follow that $M$ is permutationally equivalent to a block diagonal ...

**3**

votes

**1**answer

483 views

### Why are 1 and -1 eigenvalues of this matrix?

This is a subject I've been working on for a very long time now, but still did not manage to fully understand the interesting properties of this matrix $\mathbf{A}$.
First, let's define two matrices:
...

**1**

vote

**0**answers

48 views

### Spectral radius of a column stochastic matrix perturbed by a rank-1 matrix

$P\in \mathbb{R}^{n\times n}$ is an irreducible column stochastic matrix. $P$ is also diagonally dominant. $w \in \mathbb{R}^{n} $ is a strictly positive vector satisfying $w^T \mathbf{1} = 1$ where ...

**3**

votes

**1**answer

82 views

### Maximizing Frobenius Norm of Commutator (an opposite Procrustes problem)

I was wondering if anybody has any suggestions on the following problem:
Let $S$ be an $n\times n$ positive definite symmetric matrix. I wish to find an $n\times n$ orthogonal matrix $R$ which ...

**2**

votes

**1**answer

155 views

### Symplectic (contact) structure on $M_{n}(\mathbb{R})$

Assume that $n$ is an even number. What is a natural symplectic structure on $M_{n}(\mathbb{R})$, such that for every $1\leq k \leq n$ the manifold of $k$-rank matrices would be invariant under ...

**3**

votes

**1**answer

92 views

### Upper bounds on elements of a matrix

During my research I have come across matrices this type
$$C=B\left(B^T B\right)^{-1}B^T\ ,$$
where $B$ is an $m\times n$ real matrix. If $B^TB$ is not invertible, then $\left(B^T B\right)^{-1}$ ...

**5**

votes

**0**answers

144 views

### Characterizing matrices with rank constraint

Given matrix $M\in\Bbb Z_{\geq0,\leq b}^{n\times n}$, is there a nice method to characterize $$\mathscr{D}[M,b]=\{Q\in\Bbb R_{\geq0,\leq b}^{n\times n}:\mathsf{rank}(M-Q)= \mathsf{rank}(Q),\quad ...