All Questions
6,178 questions
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
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
views
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
views
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
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 ...
- 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
views
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
votes
5
answers
5k
views
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
votes
1
answer
708
views
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
views
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
views
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
views
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
views
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
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 ...
- 113
5
votes
6
answers
4k
views
Thorough Introduction to Singular Value Decomposition
Can you suggest a book that has a thorough introduction to Singular Value Decomposition?
- 3,613
11
votes
3
answers
2k
views
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
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, $M$, $C$, ...
- 757
5
votes
5
answers
4k
views
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
views
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
views
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
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 $...
- 11
15
votes
2
answers
3k
views
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
votes
5
answers
3k
views
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
views
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
votes
2
answers
559
views
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
votes
4
answers
1k
views
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
votes
5
answers
8k
views
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
views
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
votes
5
answers
5k
views
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
views
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
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 ...
- 6,290
-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
views
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
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
views
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
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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