All Questions
6,260 questions
3
votes
2
answers
428
views
finding an element of a vector subspace contained in the first orthant
Given a matrix $M$, I want to find a nontrivial vector in the kernel of $M$ that also lies in the first orthant, if such a vector exists. That is, I want to simultaneously solve
$$Mx = 0$$
$$x \geq 0$...
- 78
0
votes
1
answer
175
views
An inseparable lift of a regular variety.
Let $X$ be a variety over an (imperfect) field $k$, that is regular as a scheme. Let $k'/k$ be an algebraic inseparable extension (I am interested in $k'$ being the perfection or the algebraic closure ...
- 16.9k
15
votes
3
answers
4k
views
Non-diagonalizable doubly stochastic matrices
Are there constructive examples of doubly stochastic matrices (whose rows and columns all sum up to $1$ and contain only non-negative entries) that are not diagonalizable?
- 731
7
votes
2
answers
1k
views
Row reduction of sparse matrices
Let $p$ be prime (of size roughly $100$, say). Suppose that $M$ is a matrix with coefficients in $\mathbf{F}_p$ with roughly $An$ rows and $n$ columns, where $A>1$ is some fixed small constant. ...
user631
6
votes
4
answers
2k
views
The eigenvalues of the sum of two nilpotent matrices
I have a matrix that is given by $A e^{i q} + A^* e^{-i q}$ with $A$ a nilpotent $n\times n$ matrix. The eigenvalues I get turn out always to be independent of $q$ but I cannot prove it. I want to ...
- 81
2
votes
1
answer
331
views
Symmetric polynomials preserving $-1,1$ matrices
If $A$ is an $n\times n$ integer matrix, then trivially $S=A+A^t$ and $P = AA^t$
where $t$ is ``transpose", are both symmetric.
Assume that $A$ is also a "$\lbrace -1,1 \rbrace$" matrix, i.e., the ...
- 1,641
0
votes
1
answer
1k
views
Whether the system of matrix equations is always solvable
In recent days, I learned a linear algebra problem from one of my friends.
It can be stated as follows.
Given four matrices $A,B,C,D$, find three matrices $E,G,F$, not simultaneously zero, such that ...
- 1,706
2
votes
1
answer
2k
views
How to prove a unit norm matrix is the average of two unitary matrix
How to prove a unit norm matrix is the average of two unitary matrix
- 1,706
3
votes
4
answers
6k
views
Applied linear algebra textbook? [closed]
I have a copy of Linear Algebra Done Right, which I worked through years ago in college. I have been using that book to refresh my knowledge, but it does not have an applied or computational aspect ...
- 33
2
votes
2
answers
4k
views
Moore-Penrose pseudo inverse
I have an $n\times p$ matrix $Z$ with $p>n$
I have $A$, a diagonal matrix with positive entries
I would like to know if there is a known relation (as a function of $A$) between
the Moore-Penrose ...
- 21
3
votes
0
answers
528
views
A question about the generalized Lidskii-Wielandt inequality for matrices proved by Thompson and Freede
In 1971, Thomson and Freede generalized the Lidskii-Wielandt inequalites as follows (version for singular values)
Let $A$, $B$ be $n\times n$ Hermitian matrices. Suppose $\alpha_1\geq \alpha_2 \geq \...
- 227
2
votes
1
answer
205
views
Statistical estimation of singular values and vectors
My question is about the well known and well studied singular value decomposition (SVD). What I am working on right now requires performing an SVD repeatedly on a slowly varying matrix. Since I don't ...
- 111
5
votes
1
answer
3k
views
Are all topological (finite-dim) real vector spaces homeomorphic to a coordinate space?
I know that all real, finite-dimensional topological vector spaces are isomorphic to $\mathbb{R}^n$ for some $n$, but are they also homeomorphic?
The reason I'm asking this is because I was wondering ...
- 3,079
1
vote
1
answer
479
views
Is exp(rA) = (exp(A))^r for real r and A in a Banach space?
Is $e^{(rA)} = (e^{A})^r$ when $r \in \mathbb{R}$ and $A$ is an element of a Banach algebra?
Clearly if $n$ is an integer, then
$e^{(nA)} = e^{A+A \cdots +A} = e^{A}e^{A}\cdots e^{A} = (e^{A})^n$,
...
- 657
6
votes
4
answers
3k
views
Determinants of "almost identity" matrices.
Suppose that $A$ is a real square matrix with all diagonal entries $1$, all off-diagonal entries non-positive, and all column sums positive and non-zero. Does it follow that $\det(A)\neq0$? Is this ...
- 436
31
votes
4
answers
5k
views
The Frobenius morphism
I found the following list on the "Frobenius Page" by David Ben-Zvi, described by the author as "an outdated collection of intuitive ways to think about raising to the p-th power".
Generates a ...
1
vote
3
answers
2k
views
Principal curvatures and curvature directions [closed]
Last week I considered again principal curvature (pc) and principal curvature directions (pcd) of a, for the sake of simplicity, 2-manifold embedded in 3-space. In this simple case, the pc and pcd of ...
- 579
7
votes
0
answers
1k
views
Inverse of a matrix with binomial coefficients
Let $a(n,k)=(-1)^k {{2n-k}\choose k}$ for $0 \le k \le n$ and $a(n,k)=0$ else. Then it is known (cf. OEIS A005439 and A098435) that the first column of the inverse matrix of $(a(i,j))_{i,j\ge0}$ is ...
- 5,622
3
votes
2
answers
2k
views
a matrix similarity problem.
I'd like to know whether the following statement is true or not.
Let $T_1, T_2\in \mathbb{C}^{n\times n}$ be upper triangular matrices. If there exists a nonsingular matrix $P$ such that $T_1=PT_2P^{-...
- 1,858
3
votes
3
answers
2k
views
How to define the orientation of a vector space over an arbitrary field?
I know the construction of the Hodge star operator in the context of (pseudo-)euclidean real vector spaces. Apart from the scalar product it involves a orientation of the vector space, which one has ...
- 2,399
8
votes
0
answers
633
views
Can we write unitary matrices as positive linear combinations of Hermitian matrices?
The space $M_n:=M_n(\mathbb{C})$ of complex $n\times n$ matrices has the structure of a finite-dimensional complex vector space.
The space of Hermitian matrices forms a cone in this vector space $M_n$...
user2529
33
votes
2
answers
7k
views
Dimension of infinite product of vector spaces
This question is motivated by the question link text, which compares the infinite direct sum and the infinite direct product of a ring.
It is well-known that an infinite dimensional vector space is ...
- 20.8k
2
votes
1
answer
419
views
Heisenberg group over the Gaussian integers
If we take the entries of the (standard $3 \times 3$) Heisenberg group to live in the Gaussian integers $\mathbb{Z}[i]$, what is the structure of this group? Are all of its representations known?
- 1,180
57
votes
6
answers
6k
views
Is the non-triviality of the algebraic dual of an infinite-dimensional vector space equivalent to the axiom of choice?
If $V$ is given to be a vector space that is not finite-dimensional, it doesn't seem to be possible to exhibit an explicit non-zero linear functional on $V$ without further information about $V$. The ...
- 3,511
47
votes
4
answers
8k
views
Does the fact that this vector space is not isomorphic to its double-dual require choice?
Let $V$ denote the vector space of sequences of real numbers that are eventually 0, and let $W$ denote the vector space of sequences of real numbers. Given $w \in W$ and $v \in V$, we can take their "...
- 3,992
1
vote
0
answers
2k
views
Tensor Products and Intersections
Given two algebras $A$ and $B$, and two ideals $I, J \subseteq B$ with non-empty intersection, is it true that
$$
(A \otimes I) \cap (A \otimes J) = A \otimes (I \cap J)?
$$
(Where both sides of the ...
- 637
1
vote
1
answer
304
views
How do maximum norms relatively change in Euclidean translations
Let $Q$ be the cube $[-1,1]^{3}$ and $\pi$ be a plane in $\mathbb{R}^{3}$
that contains the origin but doesn't contain any vertex of $Q$. Suppose that $A$ is an invertible
linear transformation from $\...
- 11
5
votes
2
answers
2k
views
Iterated calculation of determinants
Given a $4 \times 4$ matrix $S$ over a commutative ring $R$. I want to consider it as a $2\times 2$ matrix over $M_2(R)$. Lets say $S=\left(\begin{array}{cc} A&B \\\ C&D\end{array}\right)$ ...
- 11.1k
9
votes
5
answers
2k
views
Rank of a free module without the axiom of choice
Perhaps my question is really naive. I teach this semester in algebra. I am embarassed about the proof that a free module over an integral domain has a well-defined rank. It is based on the theorem ...
- 52.4k
-3
votes
1
answer
2k
views
Eliminating redundant linear constraints? [closed]
I have an NxN matrix of linear constraints that is not of full rank. In other words, some of the constraints are linear combinations of other constraints. The "standard" linear algebra tools (...
- 159
1
vote
0
answers
265
views
"Lift and project" procedure for matrices
Definition. Let us call $n\times n$ matrix with non-negative entries good if sum of every row and column is equal to $1/n$.
Suppose we have a good matrix $A$. Let us consider the following strange "...
- 1,791
1
vote
1
answer
712
views
Sequential sampling of Gaussian and von Mises-Fisher Random Variable
I don't find any article discussing this problem, so I dare to ask it.
Suppose we are dealing with a data $x_0 \in \mathbb{R}$ and a function $f:\mathbb{R} \to \mathbb{R}$. Say we repeatedly apply $f$...
5
votes
0
answers
391
views
An operator-norm version of Siegel's Lemma
Is there a kind of Siegel's Lemma saying that if $M$ is a ``small-height'' integer matrix, then there is a "small-height" vector $x$ with $\|Mx\|=\|M\|\|x\|$? (Here $\|Mx\|$ and $\|x\|$ denote the ...
- 23k
12
votes
3
answers
2k
views
Representability of matroids over $\mathbb R$
Let $M$ be a matroid, for example viewed as being given by a finite set $X$ and a rank function $d : P(X) \to {\mathbb N}$ such that
1) $d(\varnothing)=0$, $d(\lbrace x \rbrace)=1$, for all $x \in X$,...
- 25.5k
2
votes
3
answers
28k
views
The Convergence of Jacobi and Gauss-Seidel Iteration [closed]
Hi All!
I was supposed to find a solution of Ax=b using Jacobi and Gauss-Seidel method.
The A is 100x100 symetric, positive-definite matrix and b is a vector filled with 1's.
I am iterating(k = 1,2,.....
- 31
2
votes
1
answer
646
views
Quotient by p-th roots of unity in characteristic p
Let $X$ be a variety over $k$ of characteristic $p>0$ (you can assume $k$ algebraically closed and $X$ normal) with an action of the group scheme of $p$-th roots of unity $\mu_p = {\rm Spec}\ k[\...
- 16.2k
109
votes
15
answers
12k
views
Why are matrices ubiquitous but hypermatrices rare?
I am puzzled by the amazing utility and therefore ubiquity of
two-dimensional matrices in comparison to the relative
paucity of multidimensional arrays of numbers, hypermatrices.
Of course ...
- 151k
13
votes
3
answers
2k
views
Relationship between determinants.
Given an orthogonal matrix $O$ with dimensions $4n \times 4n$ and $\det O = -1$, how to prove that
$\det[O_{11} - O_{22} + i (O_{12} + O_{21})] = 0$?
Here $O$ is a block matrix $[[O_{11}, O_{12}], [...
- 605
6
votes
0
answers
1k
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 $...
- 475
2
votes
2
answers
492
views
on existence of matrices X, Y s.t. XAY is diagonal over non-commutative ring
Given $A\in Mat_{n\times n}(R)$ where $R$ is a non-commutative associative ring are there exist any (non-zero) matrices $X, Y\in Mat_{n\times n}(R)$ such that $XAY=diag(a_1, \ldots , a_n)$ for some $...
- 1,422
9
votes
2
answers
2k
views
Basis for the Algebraic numbers over the rationals
Is there an explicit basis for the algebraic numbers as a vector space over the rationals?
- 656
2
votes
1
answer
406
views
Are these systems of linear equations always solvable
Let $X$ be a (finite) simplicial complex and let $f$ be a map from the set of its $n$-Simplices to a abelian group $A$, with the property, that every cycle maps to $0$ (extending $f$ linearly).
Let $...
- 11.1k
2
votes
0
answers
1k
views
Can we pass to the limit in Poincaré-Jaynes-Bretthorst interpolation and deconvolution?
In Science and Hypothesis, chapter XI, The calculus of probabilities, Henri Poincaré deals informally with the fundamental problem of interpolation. He concludes (see http://ia600308.us.archive.org/21/...
4
votes
1
answer
909
views
For what values of $k$ is matrix $k A - B$ positive semidefinite?
Suppose $A$ and $B$ are two $n \times n$ real symmetric matrices, and $A$ is positive semidefinite. For what values of $k \in \mathbb R$ is matrix $kA-B$ positive semidefinite (we write as $kA-B \...
- 117
0
votes
1
answer
225
views
Codimension of non-common condition is 2?
If we have n homogeneous polynomials (over algebraically closed field) $f_1\ldots , f_n$ on variables $x_0, \ldots , x_n$
$$
f_i(x_0, \ldots , x_n) = \sum_{j_0,\ldots , j_n} a_{i, j_0, \ldots , j_n} ...
- 1,422
26
votes
6
answers
14k
views
Deriving inverse of Hilbert matrix
The Hilbert matrix is the square matrix given by
$$H_{ij}=\frac{1}{i+j-1}$$
Wikipedia states that its inverse is given by
$$(H^{-1})_{ij} = (-1)^{i+j}(i+j-1) {{n+i-1}\choose{n-j}}{{n+j-1}\choose{n-...
- 1,254
32
votes
3
answers
4k
views
Example for column rank $\neq$ row rank
The proof that column rank = row rank for matrices over a field relies on the fact that the elements of a field commute. I'm looking for an easy example of a matrix over a ring for which column rank $\...
- 4,014
13
votes
4
answers
3k
views
subspaces of singular matrices
Let $A$, $B$ be square matrices over infinite field (we identify them with linear operators on the vector space of columns). It is given that for all scalars $a,b$ the matrix $aA+bB$ is singular. Does ...
- 109k
1
vote
1
answer
399
views
Which linear transformations between f.d. Hilbert spaces contract the inner product?
Given two finite-dimensional Hilbert spaces $U, V,$ a linear transformation $T:U\to V$ contracts the inner product if for all $x,y \in U,$
$$\langle x,y \rangle_U \ge \langle Tx, Ty\rangle_V.$$
...
- 1,532
6
votes
3
answers
590
views
Zariski-closed subsemigroups of SL_n(C) are groups
I would like to show that any Zariski-closed subsemigroup of $SL_n(\mathbb{C})$ is a group. If I understand correctly, this is consequence 1.2.A of http://www.heldermann-verlag.de/jlt/jlt03/BOSLAT.PDF ...
- 892