All Questions
6,026 questions
14
votes
2
answers
2k
views
Semi-linear operators
If $V_1$ and $V_2$ are finite-dimensional vector spaces over a field $E$, each equipped with an $E$-linear operator $\phi$, we can tell if $V_1$ and $V_2$ are isomorphic as $\phi$-modules by comparing ...
- 1,680
5
votes
2
answers
752
views
Is there a name for this algebraic structure?
I found myself "naturally" dealing with an object of this form:
X is a complex vector space, with a "product" (a,b) → {aba} which is quadratic in the first variable, linear in the second, and ...
- 971
4
votes
3
answers
763
views
Non-affine, projective vector field on $\mathbb{R}^n$
I wanted recently to discuss with a fairly elementary mathematics class the kinds of self-maps of Euclidean space that carry triangles to triangles. Obviously linear maps do this, and it seemed just ...
1
vote
2
answers
540
views
Using Wavelet Transforms to Approximate Matrices
It's a long time since I worked on this kind of problem, so please bear with me.
I have an approximate inverse matrix that I'm using as a preconditioner to solve the conjugate gradient method. ...
- 221
9
votes
2
answers
984
views
Spheres over rational numbers and other fields
Let K be an ordered field. Define the n-sphere:
$$S^n(K) := \{ (x_1,x_2,\dots,x_n+1) \in K^{n+1} \mid \sum_{i=1}^{n+1} x_i^2 = 1 \}$$
A set of vectors $v_1, v_2, \dots, v_r \in S^n(K)$ is ...
- 7,320
6
votes
2
answers
364
views
Algebraic characterization of transitive spaces of matrices
Fix an integer $d \ge 2$ and let $M_d$ be the space of real $d \times d$ matrices. Let $E$ be a vector subspace of $M_d$. We say that $E$ is transitive if $E \cdot \mathbb{R}^d_* = \mathbb{R}^d$, ...
- 2,479
42
votes
6
answers
12k
views
A slick proof of the Bruhat Decomposition for GL_n(k)?
On one of my exams last year, we were given a problem (we chose five or six out of eight problems) on an exam, the goal of which was to prove the Bruhat decomposition for $GL_n(k)$. I was one of the ...
- 19.6k
10
votes
3
answers
2k
views
Is $Sym^n (V^*) \cong Sym^n (V)^\ast$ naturally in positive characteristic?
Background/motivation
It is a classical fact that we have a natural isomorphism $Sym^n (V^*) \cong Sym^n (V) ^\ast$ for vector spaces $V$ over a field $k$ of characteristic 0. One way to see this is ...
- 14.7k
0
votes
3
answers
817
views
How do we construct (in a vector space) a chain of countable dimensional subspaces that can only be bounded by an subspace of uncountable dimension?
In more rigorous language:
" V: a vector space having an uncountable base
S: The set of subspaces of V that have countable dimension.
Can we construct explicitly a chain in the poset S (ordered by ...
- 45
25
votes
8
answers
15k
views
Linear Algebra Problems?
Is there any good reference for difficult problems in linear algebra? Because I keep running into easily stated linear algebra problems that I feel I should be able to solve, but don't see any obvious ...
0
votes
2
answers
253
views
Corruption and Recovery
Suppose we want to recover an input vector $f \in \textbf{R}^n$ from some measurements $y = Af + \varepsilon$. Now $A$ is an $m \times n$ matrix and $\varepsilon$ are some unknown errors. Is this ...
- 65
0
votes
2
answers
4k
views
Convergence of iterative algorithm.
For quite a long time I'm trying to prove convergence of an iterative algorithm in case of a particular system of nonlinear equations.
Here are some characteristics of this system:
It consists of n ...
2
votes
6
answers
5k
views
Finding the Square-Root of a Non-diagonalizable Positive Matrix
What methods exist for finding the square-root of a non-diagonalizabe positive complex matrix?
- 1,776
368
votes
31
answers
80k
views
Geometric interpretation of trace
This afternoon I was speaking with some graduate students in the department and we came to the following quandary;
Is there a geometric interpretation of the trace of a matrix?
This question ...
6
votes
1
answer
886
views
Linear algebra lemma
The following Lemma is in Beauville-Donagi, and I always took it for granted. Now I've tried to find a proof, but got stuck. They say it is a really simple lemma, so I may just be overlooking ...
- 14.7k
2
votes
3
answers
657
views
Are the Gell-Mann matrices extremal when used as Kraus operators for a quantum channel?
Landau and Streater proved that a set of Kraus operators, Ai, is extremal if and only if the set
$\{A_{k}^{\dagger}A_{l}\}_{k,l \ldots N}$
are linearly independent. I have seen very convincing ...
- 121
6
votes
1
answer
347
views
Sparse approximate representation of a collection of vectors
Suppose I have a collection of $n$ vectors $C \subset \mathbb{F}_2^n$. They are of course spanned by the canonical set of $n$ basis vectors.
What I would like to find is a much smaller (~ $\log n$) ...
- 583
2
votes
1
answer
148
views
an exercise on integrality of characteristic polynomials
Suppose A is a matrix with coefficient in $Q_{\ell}$, and all the coefficients of its char. polynomial are in $Z$ (thus an integral polynomial). Prove that the char. polynomial of $A^n$ is also ...
- 1,503
127
votes
4
answers
32k
views
Slick proof?: A vector space has the same dimension as its dual if and only if it is finite dimensional
A very important theorem in linear algebra that is rarely taught is:
A vector space has the same dimension as its dual if and only if it is finite dimensional.
I have seen a total of one proof of ...
7
votes
1
answer
727
views
Reference for Tate vector spaces
... aka locally linear compact vector spaces. The one reference I know is http://www.math.harvard.edu/~gaitsgde/grad_2009/SeminarNotes/Nov3-10(CentExt).pdf. Does anyone know another good reference?
- 5,548
3
votes
2
answers
2k
views
Solving a noisy set of linear equations.
Suppose we have a square $n\times n$ real matrix $A$ of full rank such that the squares of the elements in each row sum to 1, an $n\times 1$ vector of variables $x$, and an $n\times 1$ real vector $a$,...
- 41
2
votes
2
answers
1k
views
What is it called if a vector space doesn't have an additive inverse?
so, you have, for any two members of the algebraic structure A and B and any nonnegative real values a, b:
two operations: * and +, such that
a*A + b*A = (a+b)*A is in the structure
A + B = B + A ...
- 598
9
votes
2
answers
2k
views
A generalization of Boolean matrix multiplication for order-3 tensors
The Boolean matrix product of two 0-1 $n \times n$ matrices $A$ and $B$ is the matrix $C$ defined as
$$C[i,j] = \vee_{k=1}^n (A[i,k] \wedge B[k,j]).$$ If $A = B$ and the matrix is an adjacency matrix ...
- 4,729
15
votes
5
answers
18k
views
Proving "almost all matrices over C are diagonalizable".
This is an elementary question, but a little subtle so I hope it is suitable for MO.
Let $T$ be an $n \times n$ square matrix over $\mathbb{C}$.
The characteristic polynomial $T - \lambda I$ splits ...
- 7,442
3
votes
3
answers
212
views
Rank(A) and other algorithms as a polynomial
If $A = (\alpha_{ij}) \in \mathbb{C}^{nxm}$ we have simple algorithms by which to determine $\mathrm{rank}(A)$. However, is there a polynomial $f \in \mathbb{C}[\alpha_{ij}]$ where $f \colon \mathbb{C}...
- 3,165
2
votes
2
answers
3k
views
Statement of Lagrange's theorem on determinants(elementary question).
Apologies for this elementary question; but I was unable to find a reference otherwise.
Let $A, B, C$ be square matrices of the same dimension. Then,
$$\begin{vmatrix} A & C \\\ 0 & B \end{...
- 7,442
38
votes
1
answer
10k
views
Infinite tensor products
Let $A$ be a commutative ring and $M_i, i \in I$ be a infinite family of $A$-modules. Define their tensor product $\bigotimes_{i \in I} M_i$ to be a representing object of the functor of multilinear ...
- 63.1k
13
votes
8
answers
38k
views
What is the difference between matrix theory and linear algebra? [closed]
Hi,
Currently, I'm taking matrix theory, and our textbook is Strang's Linear Algebra. Besides matrix theory, which all engineers must take, there exists linear algebra I and II for math majors. What ...
- 253
1
vote
2
answers
877
views
Matrix logarithms are not unique
In my ODE class, we proved that if $\exp(L) = \exp(L')$ then the eigenvalues are congruent mod $2 \pi i$. Here, $L$ and $L'$ are two $n \times n$ matrices. I wanted to know if something more precise ...
- 22.8k
1
vote
1
answer
419
views
Is the direction of the longest line of a polytope unique?
The question pertains to a polytope that is generated by the intersection of an affine subspace with a hypercube in $p$ dimensions.
The affine subspace is given by:
$X \mbox{ u} = y$
where
$u$ &...
- 121
1
vote
1
answer
354
views
On permutation of elements of two bases of a vector space (Greub´s book)
Let {a1,a2,...,an} and {b1,b2,...,bn} be two bases for a vector space E. Fix p, 1 ≤ p ≤n. Is there a permutation σ such that
{a1,a2,...,ap,bσ(p+1),...,bσ(n)} and {bσ(1),...
3
votes
1
answer
2k
views
Conditions that allow unique solutions for Linear Diophantine equations
(This posting became very long, so I should note that there are two alternative but nearly equivalent formulations of the same question being given. The first one asks for the optimal strategy for ...
3
votes
1
answer
456
views
Standard name for basis-independent submatrices?
Given a linear map $T:H\to H$ on an inner-product space $H$ and a subspace $K\subseteq H$, define the map $T_K = \pi_K T \pi_K^* :K \to K$, where $\pi_K:H\to K$ is the orthogonal projection.
As an ...
- 11.4k
5
votes
1
answer
2k
views
Self-similar matrices? [closed]
Does anyone know anything about self-similar (infinite) matrices, with more or less fractal(-like) structure and admitting meaningful matrix-algebra operations?
- 101
4
votes
2
answers
818
views
Number of independent distances between n points in d-dimensional Euclidean space?
There are $\binom{n}{2}$ distances between $n$ points in $\mathbb{R}^d$. Not all of them can be chosen freely if $n$ exceeds the number $n_d = d + 1$. If $n = n_d$ we obviously have $\binom{d+1}{2}$ ...
- 9,720
1
vote
0
answers
393
views
iterated characteristic polynomials
If I have $N$ $M\times M$ symmetric positive definite matrices $A_i$ and an $N\times N$ positive semi-definite symmetric matrix B, let the $N\times N$ matrix $C_{ij}(\lambda)=B_{ij}$ for $i\ne j$ and $...
- 11
3
votes
4
answers
2k
views
How many parameters are needed to specify a k-dimensional subspace of R^d?
What is the number $N^d_k$ of real-valued parameters that are needed to specify a k-dimensional subspace of $\mathbb{R}^d$? And how can these parameters be interpreted?
I know: $N^d_1 = N^d_{n-1} = d ...
- 9,720
1
vote
1
answer
210
views
Extracting integer multiplicative factors from the sum of certain sets of (finite-precision) real numbers?
Update based on Michael's answer (thanks again!) - Can the LLL or PSLQ algorithms provide a (knowably - i.e. not just incidental) unique solution for the set of integer multiplicative factors? Are ...
- 43
7
votes
2
answers
1k
views
An Expectation of Cohen-Lenstra Measure
The Cohen-Lenstra measure on the set of abelian p-groups assigns $\mathbb{P}(G) = \prod_{i \geq 1} \left( 1 - \frac{1}{p^i}\right) \cdot |\mathrm{Aut}(G)|^{-1} $. Apparently, this is equivalent to ...
- 22.8k
4
votes
0
answers
306
views
Diagonalizing matrices over cyclotomic fields with unitaries
Let $F$ be a number field with a fixed embedding $F \hookrightarrow \mathbb{C}$ such that the restriction of complex conjugation from $\mathbb{C}$ to $F$ is in Gal$(F/\mathbb{Q})$ and fix a Hermitian ...
- 1,951
9
votes
9
answers
4k
views
Help me with this proof: Drop a printed map of the land on the land and there must be some common point.
Hi, I have a minor in math and this is not a homework problem - my prof mentioned it 5 years ago and I could not even begin to tackle it until I took a good intro to linear algebra (after work). ...
- 171
1
vote
2
answers
923
views
Extremum under variations of a traceless matrix
Sorry for my precedent tentative, I was a little hasty:
Ok, I think I'd better put the original problem:
I have an action of three fields: $A$ which is the spin-connection, $B$ an skew-symmetric 2-...
- 733
20
votes
8
answers
3k
views
Finitely presented sub-groups of $\operatorname{GL}(n,C)$
Here are two questions about finitely generated and finitely presented groups (FP):
Is there an example of an FP group that does not admit a homomorphism to $\operatorname{GL}(n,C)$ with trivial ...
- 28.9k
0
votes
2
answers
408
views
How to construct matrices with periodicity [closed]
Suppose I want to construct an $n\times n$ matrix ${\bf A}$ such that ${\bf A}^n={\bf I}$. Matrices that have period $n$ and admit such property are permutation matrices. However, I was wondering if ...
- 49
15
votes
3
answers
6k
views
Simultaneous diagonalization
I'm pretty sure that the following (if true) is a standard result in linear algebra but unfortunately I could not find it anywhere and even worse I'm too dumb to prove it: Let $k$ be a field, let $V$ ...
- 5,243
0
votes
2
answers
207
views
What optimization criteris should be used for this problem?
The real world version:
I have a united value (e.i. 12in, 120V 1.414 kg*m/s) where the units are specified as the rational exponents of the 5 base units; m, s, kg, C and K. Additionally, I have a set ...
- 205
23
votes
13
answers
7k
views
Pedagogical question about linear algebra
Last semester I taught a linear algebra class that is intended to introduce young students (at a sophmore-junior level) to "abstract mathematics". It seems that a major conceptual hurdle for many of ...
2
votes
1
answer
728
views
Cubic spline of a two-variable function
So, I am aware of how to (both iteratively and using a linear equation) compute the cubic spline of a one-variable function with $m$ control points. However, I am not sure how to do any type of spline ...
- 21
8
votes
2
answers
746
views
Field extension containing the eigenvectors of a Hermitian matrix
Let H be a (finite-dimensional) Hermitian matrix with algebraic numbers for its entries, all of which lie in some minimal field extension of the rational numbers; call this field ℚ(H) for short. ...
- 2,649
14
votes
3
answers
1k
views
"Conjugacy rank" of two matrices over field extension
I have posted this elsewhere and got only a partial reply. I don't know whether this qualifies the question for an open-problem tag; if it does, please anyone insert it.
Let $L$ be a field, and $K$ a ...
- 33.8k