# Tagged Questions

**2**

votes

**1**answer

78 views

### Existence of parametrizations of rational orthogonal matrices

I suppose that there are formulas which parametrize all the orthogonal matrices with rational coefficients. Does anyone know anything about it? And what are some publications that discuss this?
...

**1**

vote

**1**answer

74 views

### Neighborhood overlap matrix for a bipartite graph

Let $G$ be an undirected, simple, bipartite graph with parts $V$ (having $n$ vertices) and $W$ (having $m$ vertices). Define the following $n$-by-$n$ matrix: for any $i,j \in V$, $$a_{ij} = |N_i \cap ...

**15**

votes

**5**answers

2k views

### Origin of exact sequences

I have seen exact sequences appearing a lot in algebraic texts with different purposes. But I've never seen names of the people associated with it. Also I don't understand what's so good about showing ...

**3**

votes

**0**answers

68 views

### What do we know about the generalized eigenvalue problem involving a projector?

Consider a matrix $A\in\mathbb{R}^{n\times n}$ and a projector $P\in\mathbb{R}^{n\times n}$.
Are there results regarding the generalized eigenpairs $(v,\lambda)$ of the generalized eigenproblem
...

**3**

votes

**1**answer

98 views

### The spectral radius of a modified graph

Let $H$ be a graph and let $G=H \vee K_{1}$ be obtained by creating a new vertex and joining it to every vertex in $H$.
This situation has many different names: $G$ is called the cone or the ...

**2**

votes

**0**answers

57 views

### Can sparse matrices satisfy the Null Space Property?

Definition A matrix $A \in \mathbb{C}^{m \times N}$ with $m < N$ satisfies the Null Space Property (NSP) of order $s$ if
$$\|x_S\|_1 < \|x_{\bar{S}}\|_1, \quad \forall x \in \ker A \setminus ...

**2**

votes

**3**answers

114 views

### Applications of rank factorization or full rank decomposition [closed]

I am teaching a course on linear algebra and came to this theorem: every $m \times n$ matrix $A$ with rank $r$ admits a factorization $A = CR$ where $C$ is an $m \times r$ matrix and $R$ is an $r ...

**3**

votes

**2**answers

203 views

### Looking for a reference: double orthogonal complement in $(\mathbb{Z}/q\mathbb{Z})^n$

I'm using the following result in a computer science paper:
Let $V$ be a submodule of $(\mathbb{Z}/q\mathbb{Z})^n$ (n-tuples with addition and multiplication mod $q$). Let
$$V^\perp = \{u \in ...

**0**

votes

**0**answers

92 views

### Perturbation of spectrum and eigenspaces

Let $A \in \mathbb{C}^{n \times n}$ be an $n \times n$ matrix. Consider the rank-$1$ perturbation $A'$ of $A$ given by replacing a column $v$ of $A$ by $\alpha \cdot v$, where $\alpha \in [0, 1)$. Can ...

**3**

votes

**1**answer

502 views

### Has anybody seen my missing lemma?

I think I have a proof of the following elementary lemma (although I only need the case in which the two flags are "in general position", i.e., $F^d \cap G^i$ is minimal given the dimensions of the ...

**4**

votes

**1**answer

207 views

### The d-dimensional matrix with columns (1,0,0…), (1/2,1/2,0,…), (1/3,1/3,1/3,0,…),…, (1/d,1/d,…,1/d)

During the course of physics research on nonequilibirum statistical mechanics involving the theory of majorization, I have come across a linear transformation on a d-dimensional vector space that I ...

**2**

votes

**3**answers

363 views

### A textbook on linear algebra where involutions on linear spaces are considered

Let us call an involution on a complex linear space $X$ an arbitrary $\mathbb R$-linear map $x\in X\mapsto x^*\in X$ that satisfies the following identities:
$$
x^{**}=x,\qquad (\lambda\cdot ...

**4**

votes

**1**answer

318 views

### Rank of a 0-1-matrix

Suppose $K$ is a field of characteristic $0$. Let $M \in K^{n \times m}$ be a matrix such that every entry of $M$ is either $0$ or $1$. About this matrix, I know further that each sum over a column ...

**2**

votes

**1**answer

222 views

### Galois deformations with Panchiskin condition

Let $L/\mathbf{Q}_p$ be a finite extension and we consider a fixed $L$-linear representation $V$ of the absolute Galois group $G:=\operatorname{Gal}(\overline{\mathbf{Q}}_p/\mathbf{Q}_p)$. Assume that ...

**3**

votes

**1**answer

147 views

### What is the name of this measure of matrix “degenerateness”

Given a spanning set, consider the minimum number of vectors that you must remove in order to make it no longer span. What is this number called?
If the vectors are columns in a matrix $\Phi$, then ...

**3**

votes

**1**answer

454 views

### Diagonalize the simultaneous matrices and its background

For two $n \times n$ nonnegative definite Hermitian matrices $A$ and $B$ over the real number field $\mathbb R$:
Question1:Is there always a
nonsingular matrix $P$ over the same
field $F$ ...

**1**

vote

**1**answer

80 views

### Expected rank - computable approximations

I'm interested in finding the expected rank of some random matrix $A$ (I don't want to specify its distribution right now, since my question makes sense in general).
Computing $\mathbb{E} \ ...

**3**

votes

**1**answer

176 views

### On the divisibility of the special linear group of degree $n$ over an algebraically closed field

Let $n$ be a positive integer, $p$ a (positive rational) prime, and $\mathbb K$ an algebraically closed field. If ${\rm char}(\mathbb K) = 0$ then ${\rm GL}_n(\mathbb K)$ is divisible (see here). But ...

**16**

votes

**6**answers

1k views

### how to find/define eigenvectors as a continuous function of matrix?

I asked this (with background) here
http://stats.stackexchange.com/questions/38494/principal-component-analysis-bootstrap-and-probability-of-eigenvalue-collision
but did not really get any answers. ...

**1**

vote

**1**answer

168 views

### Decomposition of Matrix to its sub-matrix with constant rank

When we study the structure of simple graphs with a lot of $1$ or $-1$ as its adjacency eigenvalues, the rank of its adjacency matrix is very important. The reason is, in these case, we can study the ...

**3**

votes

**0**answers

119 views

### Reduction of size of orthogonal matrices.

While experimenting with orthogonal vectors I've noticed the
following transformation: If
$$
A = \begin{bmatrix}z & r \cr
c & B\end{bmatrix}
$$
is orthogonal, $z$ ...

**11**

votes

**2**answers

499 views

### Quadratic Farkas' Lemma?

The Farkas Lemma says that if a system of linear inequalities implies
yet another linear inequality, then this last inequality can be obtained by
taking a positive linear combination of the ...

**3**

votes

**2**answers

209 views

### Bounding the minimal maximum norm of a solution of a linear system.

I would be grateful for pointing me out a reference to some general bound on the $\ell_{\infty}$ norm of a solution of a linear system. To be specific, suppose that we have an underdetermined linear ...

**2**

votes

**1**answer

490 views

### Linear algebra of finite abelian groups

If $f: V \to W$ is a surjective homomorphism of vector spaces, and we have fixed a basis for $V$, it is always possible to find a basis for $W$ such that the matrix associated to $\phi$ in the two ...

**6**

votes

**1**answer

2k views

### Eigenvalues of product of two symmetric matrices

This is mostly a reference request, as this must be well known!
Let $A$ and $B$ be two real symmetric matrices, one of which is positive definite. Then it is easy to see that the product $AB$ (or ...

**5**

votes

**0**answers

402 views

### A reference on semisimple linear algebra

Is there any literature where the tools familiar from (multi)linear algebra are systematically transferred to the setting of semisimple modules over noncommutative rings?
In fact this question is a ...

**0**

votes

**0**answers

156 views

### Changing basis on an extension of a free Z-module.

Consider a finite-rank free $Z$-module $Y$. Let $c: Y \times Y \rightarrow Z$ be a $Z$-bilinear form. Assume that $c(y_1, y_2) + c(y_2, y_1)$ is even, for all $y_1, y_2 \in $. Then $c$ ...

**2**

votes

**3**answers

386 views

### On certain decomposition of unitary symmetric matrices

This is by any means elementary, but since I have asked this question on Stark Exchange but received no satisfactory answers I decide to post it here.
It is well known that a symmetric matrix over ...

**26**

votes

**1**answer

947 views

### solving linear equations made difficult

(Note: This is a what's-in-the-literature question, not a what's-mathematically-true question, but I believe both are considered valid kinds of MathOverflow question.)
I saw this amusing derivation ...

**2**

votes

**4**answers

311 views

### A polynomial homomorphism from Gl to the group of units is a power of the determinant

I was browsing MO and stumbled upon this post, and I got very curious. I searched for about half an hour and could not find a proof for the statement that any polynomial group homomorphism ...

**5**

votes

**0**answers

124 views

### reference for perturbation of projection result

Let $A$ and $B$ have the same rank and dimensions. If $P_A$ denotes the projection onto the range space of $A$, then
$$
\|P_A - P_B\|_2 \leq \|A - B\| \cdot \min (\|A^\dagger\|_2, \|B^\dagger\|_2).
$$
...

**3**

votes

**4**answers

758 views

### determinants and polynomials in matrices

Muirhead (1982, "Aspects of Multivariate Statistical Theory") references on page 59
a result (from MacDuffee, 1943, chap 3, "Vectors and Matrices") a book I cannot find):
" The only polynomials in ...

**1**

vote

**0**answers

108 views

### Counting equivalence classes in the transitive closure of two equivalence relations

Let $X$ be a finite set, and let $P_i$ and $Q_j$ be two partitions of $X$:
$$\bigsqcup_i P_i = \bigsqcup_j Q_j = X.$$
The finest partition which is nevertheless coarser than both $P$ and $Q$ is ...

**5**

votes

**2**answers

429 views

### Unpublished work of Wielandt

Wielandt wrote a paper titled "Remarks on diagonable matrices".
According to Mathematische Werke - Mathematical Works : Linear Algebra and Analysis
by Helmut Wielandt, Hans Schneider, Bertram Huppert ...

**4**

votes

**2**answers

397 views

### signs of eigenvalues of quadratic form

Let $A=(a_{ij})_{i,j=1}^n$ be a symmetric real matrix, $M_k:=det(a_{ij})_{1\leq i,j\leq k}$ be its minors and $M_k\ne 0$ for all $k$. Then signs of eigenvalues of $A$ are equal (up to some ...

**2**

votes

**1**answer

137 views

### 'Compute' Integral equivalence of matrices

Hi.
For a matrix $D \in \mathbb{Z}^{n \times n}$ and a symmetric, positive definite integral even matrix $S \in \mathbb{Z}^{n \times n}$ put $S[D] := D^TSD$ where the $\cdot^T$ means 'transposed'. ...

**4**

votes

**1**answer

251 views

### Spectral Properties of $A(I-A)^{-1}$

I am working with a class of matrices $A$ which are non-negative-definite, not symmetric, and have maximum eigenvalue less than 1. I am interested in the spectral properties of the matrix $H = A(I - ...

**1**

vote

**1**answer

186 views

### references for families of conditionaly negative definite matrices

We say that a matrix $A\in M_n(\mathbb{C})$ is a conditionaly negative definite matrix if it is hermitian and if for all complex numbers $c_1,\ldots,c_n$ such that $c_1+\cdots +c_n=0$ we have
$$
...

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votes

**0**answers

137 views

### Slices of Simplices that are Simplices, Reference?

I am trying to find a reference for the following fact. It is elementary and not hard to prove, but I haven't been able to find the question treated anywhere.
Let $A$ be an $l\times n$ matrix with ...

**1**

vote

**1**answer

495 views

### Characterizing the set of self-orthogonal complex vectors

Let $v\in \mathbb{C}^n$ be an $n$ dimensional complex vector. Define the non-standard bilinear form $\left< u,v \right> = u^T v$ (the usual inner product except without the conjugation). What ...

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votes

**1**answer

526 views

### Calculating a curvature tensor by polarization

I'm reading some articles by Siu and Nannicini on the Weil-Petersson metric associated to families of compact Kahler-Einstein manifolds. In each article Siu and Nannicini construct a Weil-Petersson ...

**3**

votes

**3**answers

472 views

### Is this statement about the real edge space of a graph known or trivial?

The statement is:
($u$ is a fixed node in a fixed graph $G$)
$G$ is 3-connected
if and only if
the set of u-cycles span $\mathbb{R}^{E(G)}$.
A u-cycle is a simple (no vertex repetitions) cycle in G ...

**5**

votes

**2**answers

417 views

### adjoint of multiplication operator in a commutative algebra

Dan Popescu asked me the following question, and since I'm not an expert I'm throwing his question on MO.
Suppose that $A$ is a finite-dimensional vector space over an ordered field $k$ with ...

**1**

vote

**0**answers

170 views

### What is the MP pseudoinverse's role in statistical learning and Self-Organizing Maps?

During a discussion in our lab last month, a professor mentioned to me that the behavior of Self-Organizing Maps can be described in terms of repeated applications of the Moore-Penrose psuedoinverse, ...

**6**

votes

**1**answer

405 views

### Reference for a formula expressing the characteristic polynomial of a sum of endomorphisms

Let $R$ be a ring, $A$ a (not necessarily commutative) $R$-algebra and $M$ a (left) A-module which is free of finite rank as an $R$-module. If $a\in A$ then multiplication with $a$ on $M$ is an ...

**0**

votes

**2**answers

450 views

### A basis for $\mathbb{Q_p}$ as a vector space over $\mathbb{Q}$

I was looking for a reference that illustrates a $\mathbb{Q}$-vector space basis for the field of p-adic numbers under the following action. Given a rational number $q$. write, $q=\frac{m}{n}$ where ...

**2**

votes

**1**answer

418 views

### Largest eigenvalue of a periodic Jacobi matrix

There is a vast literature on Jacobi matrices, I just don't know where to start looking. I'm interested in estimating the largest eigenvalue of the $n\times n$ periodic Jacobi matrix $D+P+P^{-1}$, ...

**0**

votes

**2**answers

620 views

### Fast algorithms for computing nullspace of a positive semidefinite matrix over Z

Let $A \in \mathbb{Z}^{n \times n}$ be a positive semidefinite sparse matrix. I am looking for asymptotically fast algorithms for computing the nullspace basis of $A$ (or just random elements in the ...

**8**

votes

**1**answer

165 views

### Operator compression preserving lowest energy eigenspace.

I have a large ($10^6$ by $10^6$) sparse ($0.4$% nonzero) hermitian matrix $H$ arising from the discretization of an elliptic PDE. I would like to approximate $H$ with a smaller matrix $H'$ in such a ...

**3**

votes

**0**answers

538 views

### Generalized Courant-Fischer theorem

Consider some quaternionic matrix $A$. A right eigvenvalue of $A$ is a quaternion $q$ such that $Ax=xq$ for some $x\in \mathbb{H}^n$. Similarly, a left eigenvalue of $A$ is quaternion $q$ such that ...