All Questions
6,289 questions
2
votes
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 ...
- 54.6k
8
votes
2
answers
2k
views
Hermitian matrices with prescribed number of positive and negative eigenvalues
Let $H$ be a linear subspace of the space of Hermitian $n\times n$ matrices. Is there a good characterization of those $H$ such that every $A\in H$ has at least $k$ positive and $k$ negative ...
- 971
31
votes
10
answers
9k
views
When to pick a basis?
Picking a specific basis is often looked upon with disdain when making statements that are about basis independent quantities. For example, one might define the trace of a matrix to be the sum of the ...
- 2,649
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
91
votes
5
answers
124k
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Eigenvalues of matrix sums
Is there a relationship between the eigenvalues of individual matrices and the eigenvalues of their sum? What about the special case when the matrices are Hermitian and positive definite?
I am ...
- 1,013
2
votes
4
answers
5k
views
What are the components of a transpose operator from $\mathbb R^{n\times n}$ to $\mathbb R^{n\times n}$?
Say I'm working in the space of linear transformations from $\mathbb R^n$ to $\mathbb R^n$ and I've picked a basis so I can identify with any operator a component matrix in $\mathbb R^{n\times n}$. ...
- 123
2
votes
2
answers
4k
views
Shear transformations
Where can I learn more about shear matrices?
The Wikipedia article is not enough, and sadly it does not have any references.
I understand they are linear transformations. Do they form a group? How ...
1
vote
8
answers
2k
views
Bivectors in 3 and 4 dimensions
The big questions behind are:
Is a bivector a two-form?
Why a bivector is simply a vector in 3 dimensions?
How to distinguish between vectors and bivectors in 3D?
Why all bivectors are not vectors ...
- 19
30
votes
7
answers
4k
views
When is a monic integer polynomial the characteristic polynomial of a non-negative integer matrix?
Suppose $P(x)$ is a monic integer polynomial with roots $r_1, ... r_n$ such that $p_k = r_1^k + ... + r_n^k$ is a non-negative integer for all positive integers $k$. Is $P(x)$ necessarily the ...
- 118k
1
vote
1
answer
679
views
Quantifying aggregate vector strength/vector arithmetic
Say I have 5 vectors and I measure the similarity of each one to a fixed reference vector using cosine similarity. But now what I want to do is understand the aggregate or collective strength of these ...
- 11
24
votes
5
answers
6k
views
Generators for congruence subgroups of SL_2
For positive integers $n$ and $L$, denote by $SL_n(Z,L)$ the level $L$ congruence subgroup of $SL_n(Z)$, i.e. the kernel of the homomorphism $SL_n(Z)\rightarrow SL_n(Z/LZ)$.
For $n$ at least $3$, it ...
- 44.8k
12
votes
2
answers
828
views
Matrices into path algebras
I was thinking about quivers recently, and the following idea came to me.
Let ei,j denote the matrix unit in Mn for 1 ≤ i,j ≤ n. Let Γ denote the complete quiver on vertices {1, …,...
- 1,756
7
votes
2
answers
1k
views
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 ...
- 54.6k
18
votes
3
answers
2k
views
Elementary $\mathrm{Ext}^1$ intuition
$\DeclareMathOperator{\Hom}{\operatorname{Hom}}\DeclareMathOperator{\Ext}{\operatorname{Ext}}$I am wondering what sort of basic basic intuitive meaning $\Ext^1(M,N)$ has.
As a base case: if $M$ and $N$...
- 411
9
votes
8
answers
6k
views
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 ...
- 7,800
2
votes
1
answer
821
views
Question about orthogonal matching pursuit
Let y be a n-vector, X a n-by-p matrix of full rank (p < n) and b a p-vector, so that y = Xb + e, for some noise vector e. I am not sure how to show reduction of error in orthogonal matching ...
- 151
19
votes
4
answers
2k
views
Variation on a matrix game
The original problem appeared on last year's Putnam exam:
"Alan and Barbara play a game in which they take turns filling entries of an initially empty 2008×2008 array. Alan plays first. At each turn, ...
- 1,975
22
votes
2
answers
14k
views
Infinite matrices and the concept of "determinant"
Suppose we have an infinite matrix A = (aij) (i, j positive integers). What is the "right" definition of determinant of such a matrix? (Or does such a notion even exist?) Of course, I don't ...
- 1,861
9
votes
2
answers
868
views
Orbits of real groups, canonical forms of matrices
There are a lot of results in textbooks concerned with canonical forms of matrices under certain complex groups of transformations, e.g. GL(n|C), O(n|C),...
Could anybody give me references where the ...
- 93
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 ...
- 44.8k
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?
- 41
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 ...
- 1,069
0
votes
1
answer
336
views
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)...
- 11
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
1
vote
2
answers
2k
views
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 ...
- 131
4
votes
3
answers
2k
views
Conjugation in SU(2)
For any two matrices $P,Q \in SU(2)$, with $tr(P)=tr(Q)=0$, does there always exist some $G\in SU(2)$ such that $G P G^{-1} = -P$, and $G Q G^{-1} = -Q\ ?$
- 1,129
11
votes
1
answer
410
views
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
5
votes
2
answers
702
views
Linear Algebra Over $F_{2}$
Suppose we call a subset S of $F^{n}$ ($F$ is the field with two elements) good if for any $x$ and $y$ (possibly $x=y$) we have $[x,y]=1$ where $[ , ]$ denotes the obvious bilinear form on F. What's ...
10
votes
5
answers
990
views
Non-conjugate words with the same trace
Let n>=2, p a large prime, G = SL_n(Z/pZ).
If n=2, there are words that, while not conjugate in the free group, do have identical trace in G. For example, tr(g h^2 g^2 h)= tr(g^2 h^2 g h) for all g, ...
- 20.2k
40
votes
6
answers
6k
views
Linear transformation that preserves the determinant
It seems "common knowledge" that the following holds:
Let $T$ be a linear transformation on $n\times n$ matrices with complex coefficients that preserves the determinant. Then there exists ...
- 511
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, ...
- 1,069
2
votes
1
answer
925
views
Theta Functions and Cousins
So I am (barely) familiar with the construction of the theta function of an integral lattice $L$. The theta function, as I understand it, is defined as the function which takes a variable $z$ and ...
- 27.5k
1
vote
1
answer
322
views
Request for info on the space of commuting matrices preserving a flag.
Fix a flag of subspaces V1 in V2 in V3, etc. all in Cn.
Consider the space of pairs of commuting linear transformations A and B such that:
A preserves the flag (i.e. A(Vi) is in Vi), and
B strictly ...
- 44.7k
43
votes
6
answers
9k
views
"A gentleman never chooses a basis."
Around these parts, the aphorism "A gentleman never chooses a basis," has become popular.
Question. Is there a gentlemanly way to prove that the natural map from $V$ to $V^{**}$ is surjective if $V$...
- 5,275
7
votes
2
answers
2k
views
What's the correct notion of determinant of a bilinear pairing?
By a pairing on a vector space $V$, I mean a linear map $A : V \otimes V \to R$. If $V$ is $n$-dimensional ($n < \infty$), then I can define the determinant of $A$ by considering the canonical ...
- 54.6k
18
votes
2
answers
3k
views
Zeta-function regularization of determinants and traces
The short answer to my question may be a pointer to the right text. I will give all the background I know, and then ask my questions in list form.
Let A be an operator (on an infinite-dimensional ...
- 54.6k
35
votes
5
answers
13k
views
Linearity of the inner product using the parallelogram law
A norm on a vector space comes from an inner product if and only if it satisfies the parallelogram law. Given such a norm, one can reconstruct the inner product via the formula:
$2\langle u,v\rangle ...
- 26.8k
9
votes
6
answers
8k
views
How to approximate a solution to a matrix equation? [closed]
Suppose a matrix equation $Ax = b$ has no solution ($b$ is not in the column space of $A$)
How can I find a vector $x^\prime$ so that $Ax^\prime$ is the closest possible vector to $b$?
- 461
62
votes
9
answers
23k
views
Can a vector space over an infinite field be a finite union of proper subspaces?
Can a (possibly infinite-dimensional) vector space ever be a finite union of proper subspaces?
If the ground field is finite, then any finite-dimensional vector space is finite as a set, so there are ...
