All Questions
6,289 questions
1
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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
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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
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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
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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
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8
answers
3k
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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
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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 ...
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15
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3
answers
6k
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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
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13
answers
7k
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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
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1
answer
728
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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
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3
answers
1k
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"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
5
votes
1
answer
2k
views
Inverting a covariance matrix numerically stable
Given an $n\times n$ covariance matrix $C$ where $n$ around $250$, I need to calculate $x\cdot C^{-1}\cdot x^t$ for many vectors $x \in \mathbb{R}^n$ (the problem comes from approximating noise by an $...
- 51
3
votes
4
answers
3k
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spectral radius of a matrix as one element changes
Here's my question --
Let $A$ be an $n \times n$ real matrix, and suppose that the spectral radius $\rho(A)$ is less than one (spectral radius = max eigenvalue). Let's choose some $1 \leq i \leq N$ ...
- 147
11
votes
3
answers
500
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Local-globalism for similar matrices?
My background on number theory is very weak, so please bear with me...
Given two matrices $A$ and $B$ in $\mathbb{Z}^{n\times n}$. Assume that for every prime $p$, the images of $A$ and $B$ in $\...
- 33.8k
4
votes
1
answer
720
views
"Transient" of the discrete-time Riccati equation
It is a well-known result that, if the pair $(A,Q^{1/2})$ is stabilizable and the pair $(A, C)$ is detectable, the solution to the discrete-time Riccati recursion
$P(t+1) = A P(t) A^T - A P(t) C^T\ (...
- 141
8
votes
4
answers
748
views
Tensored Over Abelian Groups?
Suppose I have an category additive category C (i.e. the hom sets are enriched in abelian groups and there are finite direct sums). Suppose further that C has cokernels. Then I can make C tensored ...
- 27.5k
4
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3
answers
323
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Approximately known matrix
What linear algebraic quantities can be calculated precisely for a nonsingular matrix whose entries are only approximately known (say, entries in the matrix are all huge numbers, known up to an ...
- 22.1k
3
votes
2
answers
536
views
Broken Symmetry
I have a tangled web of ideas about natural transformations, vector spaces, equivalence classes, local coordinates, etc. in my head that I'm trying to unravel. So here are some of the questions I ...
5
votes
3
answers
3k
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Endomorphisms of vector bundles
I'm a bit stuck, and I'm hoping someone can help me out. I have a vector bundle $E$ on an algebraic curve (the ones I am interested in are holomorphic, but I'm sure that doesn't matter so much for ...
- 53
4
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5
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5k
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conjugate gradient iteration
I'm having problems understanding why the conjugate gradient method breaks down for singular matrices. I've read a good introduction to intuitively understanding the CG method through visualizing the ...
- 51
9
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1
answer
708
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Hilbert spaces are induced by a bilinear form. How about n-linear forms?
A Hilbert space is a complete vector space equipped with scalar product, i.e. a symmetric positive definite bilinear form.
What if we replace 'bilinear' by 'n-linear'? One might wonder, whether the $...
- 5,327
2
votes
3
answers
946
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How can I measure the Morse index in infinite dimensions?
Let $V$ be a vector space over $\mathbb R$, and $a: V\otimes V\to \mathbb R$ a symmetric bilinear pairing. Recall that the Morse index of $a$ is the maximal dimension of any subspace $V_- \subseteq V$...
- 54.6k
2
votes
4
answers
3k
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Splitting a space into positive and negative parts
Let $V$ be a vector space over $\mathbb R$. A symmetric bilinear pairing on $V$ is a linear map $a: V\otimes V \to \mathbb R$. Because $\mathbb R$ is characteristic not-two, I will freely confuse ...
- 54.6k
16
votes
3
answers
3k
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A riddle about zeros, ones and minus-ones
I was asked this years ago, but I don't remember by whom, and have never managed to solve it.
Consider the $2^n \times n$ matrix of all vectors in {-1,1}$^n$.
Someone comes and maliciously replaces ...
- 435
9
votes
6
answers
4k
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Exact short sequences of vector spaces
If possible, how could one prove that every short exact sequence $0 \to A \xrightarrow f B \xrightarrow g C \to 0$ of vector spaces (here $A$, $B$ and $C$) splits without using any basis of $A$, $B$ ...
- 117
11
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1
answer
4k
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Singular value decomposition over finite fields?
What is the definition of a singular value over a finite field $\mathcal{F}$ of a matrix ${\bf A}$ in $\mathcal{F}^{m\times n}$? Is there a geometric intuition in the same manner as with the real case ...
- 113
5
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6
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4k
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Thorough Introduction to Singular Value Decomposition
Can you suggest a book that has a thorough introduction to Singular Value Decomposition?
- 3,613
11
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3
answers
2k
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Matrices whose nullspace is nicely shaped
I'm looking for natural conditions on $a_{ij}$ to guarantee that the null space of the $n\times m$ matrix $A=(a_{ij})$ has a nice basis.
The null space of { {1,-2,1,0,0}, {0,1,-2,1,0}, {0,0,1,-2,1} } ...
- 9,756
5
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1
answer
363
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Under what conditions do eigenvalues of a quadratic eigenvalue problem come in reciprocal pairs?
Suppose we have a quadratic eigenvalue problem $\lambda^2 M + \lambda C + K$. Under what conditions is the following statement true: If $\lambda$ is an eigenvalue, so is $1/\lambda$?
Here, $M$, $C$, ...
- 757
5
votes
5
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4k
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A random walk matrix has eigenvalue 1 with multiplicty 1 - why?
A random walk matrix has largest eigenvalue 1 with multiplicty 1 - why?
Let $G$ be a non-directed, regular connected graph with degree $d$. Let $A$ be its random walk matrix, i.e. it's adjacency ...
- 5,327
33
votes
4
answers
10k
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Definition of inner product for vector spaces over arbitrary fields
Is there a canonical definition of the concept of inner products for vector spaces over arbitrary fields, i.e. other fields than $\mathbb R$ or $\mathbb C$?
- 341
4
votes
1
answer
173
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factorization of the product of a matrix element and its cofactor
Hi,
this is kind of continuation of this thread to concentrate on a specific problem from linear algebra and analysis that, I think, is rather interesting for itself.
Here we go:
1) Main problem: ...
- 7,127
3
votes
1
answer
263
views
Asymptotically multiplicative functions and matrices
Hi,
Let $\mathbb{N}_{cop}^2$ denote the set of all pairs of coprime natural numbers. A function $f:\mathbb{C}\rightarrow\mathbb{C}$ is called asymptotically multiplicative, iff $\epsilon_{m,n}:=f(mn)...
- 7,127
5
votes
1
answer
603
views
Hermite normal form in families
How does Hermite normal form (over $Z$) vary in families? I.e. if I have an $n\times m$ matrix $M$ whose entries are integral polynomials in some integral variable $x$, how does the Hermite normal ...
- 2,562
1
vote
1
answer
2k
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Real and Complex Projections
A projection $P$ on a real vector space is defined to be a linear mapping such that $P^2 = P$. For projections on complex vector spaces why does one require the extra condition that $P^* = P$, where $...
- 11
15
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2
answers
3k
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How to compute the rank of a matrix?
Okay, that's a misleading title. This is a somewhat subtler problem than undergraduate linear algebra, although I suspect there's still an easy answer. But I couldn't resist :D.
Here's the actual ...
- 12.6k
7
votes
4
answers
2k
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Is there a name for the matrix equation A X B + B X A + C X C = D?
I happen to be working on a problem that reduces to solving the following equation:
$$\mathbf{A X B} + \mathbf{B X A} + \mathbf{C X C} = \mathbf{D}$$
where A through D are known matrices ( A, B, D ...
- 1,890
12
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5
answers
3k
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How can I learn about doing linear algebra with trace diagrams?
There is a wikipedia article. There is a paper by Elisha Peterson. I tried reading these but they don't seem to click for me.
Are there books or other resources for learning how to do linear algebra ...
- 3,613
7
votes
8
answers
1k
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Ways to Synthesize Topics in Linear Algebra
Hello, I am currently studying linear algebra right now. In general, the material is pretty straight-forward but it doesn't seem particularly interesting. I suppose that the main thing that I am ...
- 317
15
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2
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559
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Which quadratic forms on $\Lambda^2 V$ come from quadratic forms on $V$?
Let $V$ be a finite dimensional vector space, say over $\mathbf R$. Let $g \in S^2 V^*$ be a quadratic form on $V$. Then $g$ induces a quadratic form $\Lambda^2 g \in S^2 \Lambda^2 V^*$ on $\Lambda^...
- 151
3
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4
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1k
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How can I generate (suitably random) symplectic matrices?
I would like to write a computer script to generate a lot of symplectic matrices. How can I do this? Is there a parameterization of all symplectic matrices?
- 481
16
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5
answers
8k
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Which graphs have incidence matrices of full rank?
This is a follow-up to a previous question. What graphs have incidence matrices of full rank?
Obvious members of the class: complete graphs.
Obvious counterexamples: Graph with more than two ...
- 1,890
8
votes
1
answer
638
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Composite residues with determinant denominators
I am looking for a good reference on composite residues of multi-variable contour integrals (something better and more explicit than Griffiths and Harris or Tsikh). This means I want to evaluate $\...
- 81
5
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5
answers
5k
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Notions of Matrix Differentiation
There are a few standard notions of matrix derivatives, e.g.
If f is a function defined on the entries of a matrix A, then one can talk about the matrix of partial derivatives of f.
If the entries of ...
- 590
2
votes
2
answers
356
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Is there a specific name for matrices with nonsingular principal submatrices?
Is there a specific name for matrices with nonsingular principal submatrices?
- 1,638
3
votes
1
answer
2k
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How do you construct a symplectic basis on a lattice?
Is this possible to do constructively? The only sources that I have for the possibility of this construction is an exercise in Lang's Algebra (on p. 598, I believe) which states that one can be ...
- 6,290
-2
votes
1
answer
162
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What is the weakest condition on the matrices A_k that guarantees v_k->0 => A_kv_k->0 ? [closed]
What is the weakest condition on the sequence of real matrices A_k that guarantees that whenever a sequence of real vectores v_k converges to zero, the product A_kv_k also converges to zero?
Edit: ...
- 1,638
7
votes
1
answer
2k
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Graphs with incidence matrices whose pseudoinverses are proportional to their transposes
When I was working on my PhD dissertation, I came across a physical situation involving nodes and flows between them. It turned out that I was working with a complete oriented graph $K_n$ (all nodes ...
- 1,890