All Questions
5,882 questions
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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 ...
- 2,649
5
votes
1
answer
2k
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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
2
votes
2
answers
356
views
Is there a specific name for matrices with nonsingular principal submatrices?
Is there a specific name for matrices with nonsingular principal submatrices?
- 7,329
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\ (...
- 4,577
4
votes
3
answers
323
views
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 ...
- 1
11
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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 $\...
- 156k
11
votes
1
answer
410
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An "existence contra partition of unity" statement for integer matrices?
While reading a blog post on partitions of unity at the Secret Blogging Seminar the following question came into my mind.
Let $n$ be a positive integer and let $B_1$ and $B_2$ be $n \times n$ ...
- 2,600
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 ...
- 26.8k
3
votes
2
answers
536
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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 ...
CommunityBot
- 1
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 ...
- 27.5k
9
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8
answers
6k
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Which computer algebra system should I be using to solve large systems of sparse linear equations over a number field?
This is related to Noah's recent question about solving quadratics in a number field, but about an even earlier and easier step.
Suppose I have a huge system of linear equations, say ~10^6 equations ...
CommunityBot
- 1
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 ...
- 56.6k
15
votes
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 ...
- 2,416
11
votes
1
answer
4k
views
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 ...
- 297
5
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5
answers
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 ...
- 297
5
votes
1
answer
363
views
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, $M$, $C$, ...
- 228
3
votes
1
answer
2k
views
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 ...
- 156k
21
votes
4
answers
4k
views
Rings over which every module is free
We know that modules over skewfields are free. Is the converse true? In other words, is it true that a nontrivial ring over which every module is free is a skewfield?
If the ring A is commutative, ...
- 44.7k
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)...
- 25.8k
1
vote
1
answer
2k
views
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 $...
- 5,992
2
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3
answers
493
views
In an n-dimensional linear 2nd-order ODE, why is the transpose-inverse to a system of solutions also a solution?
I'm at a sticky spot in my research. Namely, I have a particular fact, and it ought to have a short proof, but the only way I know how to show it is long and drawn out, and I don't like it and worry ...
- 3,322
5
votes
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 ...
- 2,284
7
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2
answers
1k
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Abelianization of Lie groups
If G is a group, its abelianization is the abelian group A and the map G → A such that any map G → B with B abelian factors through A. Abelianization is a functor, and in general a very ...
CommunityBot
- 1
-2
votes
1
answer
162
views
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 ...
- 5,992
1
vote
4
answers
385
views
Is it that only with normal matrices, the transition matrix to its [del: inherent] [ins: own] basis is unitary?
Does this even make sense what I translated into english?
PS. I am probably gonna delete this question eventually
- 13
2
votes
2
answers
454
views
Is the center of a free (as a module) algebra free?
A submodule of a free module need not be free (for instance, in the free Z[X]-module Z[X] the submodule generated by 2 and X is not free). But over a principal ideal domain, submodules of free modules ...
- 53.3k
8
votes
1
answer
572
views
Lifting bases for (Z/pZ)^n to Z^n
The following question came up in my research. I suspect that it has a slick answer,
but I can't seem to find it.
Fix an integer n>=2 and a prime p. Define X(n) to be the set of primitive
vectors ...
- 6,579
11
votes
2
answers
1k
views
What is the size of the category of finite dimensional F_q vector spaces?
The size of a finite skeletal category C in the sense of Leinster is defined as follows: Label the objects of C by integers 1,2,...,n and let aij be the number of morphisms from i to j (for i and j ...
- 2,600
4
votes
1
answer
714
views
How do you rotate a matrix to maximum sparsity?
Given a matrix M, I want to find an orthogonal matrix U that maximizes the number of entries that are zero in the product MU. How do I go about doing this?
- 5,548
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1
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336
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Change of basis with Multilinear fucntion [closed]
Take a multi-linear function(or functional) M that takes m arguments V1…Vm, each with a dimension n. Consider only the case where m=n. Let there be a change of basis performed on the arguments(V1...Vm)...
CommunityBot
- 1
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2
answers
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Friedberg, Insel, and Spence Linear Algebra example
In the chapter 6.4 on normal and self-adjoint operators, there is an example of an infinite dimensional inner product space H that has a normal operator but that has no eigenvectors.
The space is the ...
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