All Questions
6,287 questions
5
votes
4
answers
2k
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Diagonalization of Infinite Hermitian matrices
We know that $n\times n$ square Hermitian matrices can be diagonalized and have real eigenvalues.
Suppose I have a countable sized Hermitian matrix $A=(a_{ij})$ where the indices $i$ and $j$ run ...
user2529
5
votes
1
answer
1k
views
Techniques for lower-bounding angle between two eigenvectors of a matrix
Are there any techniques for lower-bounding the angle between eigenvectors of a matrix? Or a lower bound on the related quantity of the condition number of the matrix of eigenvectors? In particular I'...
- 303
1
vote
1
answer
546
views
A 3*3 matrix space problem
A matrix subspace $S\subset M_n(C)$ is called "good", if there is two linear independent elements of $S$, says $E_1,E_2$ which are simultaneously singular valued decomposable, i.e., $E_1=UD_1V$ and $...
- 1,503
10
votes
1
answer
1k
views
Bounds on $\|P^{k+1} - P^k\|$ for $n$ by $n$ stochastic matrix $P$ with trace $n-1$ and integer $k\gg n$
The problem:
We have a $n$-state Markov chain with arbitrary initial distribution and transition matrix $P$ that is arbitrary except that we know that $P$ has trace $n-1$. Of course $P$ is also a ...
- 303
6
votes
2
answers
2k
views
Tight bound for sum of entries of the inverse of a nonnegative matrix
While playing around with certain non-negative matrices, I got stuck at the following question.
Let $A$ be a strictly positive-definite $n \times n$ matrix ($n \ge 3$), with ones on the diagonal, and ...
- 28.6k
2
votes
1
answer
1k
views
The difficulty of generate complex Hadamard matrix
A complex $n\times n$ matrix $A=[a_{ij}]$ is called a Hadamard matrix if $A^{+}A=nI$ and $|a_{ij}|=1$ holds for all $i,j$, where $A^{+}$ denotes the conjugate transposed matrix of $A$, and a vector $...
- 1,503
9
votes
0
answers
1k
views
coordinate-free proof of transitivity of norms or traces
Hello:
Suppose $A$ is a finite free $B$-algebra and $B$ is a finite free $C$-algebra. Does anyone
know a coordinate-free proof (i.e. without choosing bases) of the identity:
$N_{A/C} = N_{B/C}\circ ...
- 647
8
votes
0
answers
221
views
Standard polynomials applied to matrices (bis)
The standard polynomial in $r$ non-commuting indeterminates $x_1,\ldots,x_r$ is defined by
$${\mathcal S}_r(x_1,\ldots,x_r):=\sum_{\sigma\in S_r}\epsilon(\sigma)x_{\sigma(1)}x_{\sigma(2)}\cdots x_{\...
- 52.3k
6
votes
1
answer
632
views
Norm of commutators (bis)
This question is slightly related to a popular one with the same title (see here).
Let $k$ be a field with characteristic zero. It is known (see Exercise 310) that a matrix $A\in M_n(k)$ is nilpotent ...
- 52.3k
1
vote
0
answers
169
views
Sum of two free o-submodules in a vector space over a local field
Let $V$ be a countably infinite dimensional $K$-vector space over a local field $K$ (nontrivially discretely valued with finite residue field). Let $o$ be the ring of integers of $K$.
Given two free ...
- 107
3
votes
3
answers
346
views
Computational solutions to families of systems of linear equations
Question
Does there exist a computer package that will solve families of systems of linear equations over a field of prime characteristic?
An Example
Suppose I wanted to know when the following ...
- 136
1
vote
0
answers
221
views
Nonunique low-rank matrix completion from a few entries
Suppose we want to have a good approximation for the following NP-hard problem
$$\min_{\bf X} \operatorname{rank}({\bf X}) \text{ s.t. } \mathcal{A}({\bf X}) = {\bf b}, {\bf X} \succeq 0$$
where ${\bf ...
- 449
19
votes
1
answer
904
views
Is the norm of a $0-1$ matrix (almost) attained on a $0-1$ vector?
I'd like to state explicitly a problem which was somehow hidden in my three-week-old post:
Does there exist an absolute constant $c>0$ with the property that for any matrix $M\in{\mathcal M}_{m\...
- 23k
1
vote
4
answers
946
views
Representation of Lie algebra sl_2.
Consider the Lie algebra $sl_2$
with the standard basis $(e,f,h),$ where
\begin{equation*}\label{sl2}
[h,e]=2\,e, [h,f]=-2\,f,[e,f]=h.
\end{equation*}
Let $V$ be finite-dimensional $sl_2$-...
- 301
4
votes
1
answer
314
views
For which linear endomorphisms can one find a basis such that the matrix is nonnegative?
Hi there,
Consider linear endomorphisms ("endos") of a finite dimensional vector space.
How can those endos be characterized, for which said vector space has a basis with respect to which the endo ...
- 103
5
votes
0
answers
530
views
Given two linear operators A and B over a finite field, is there a third operator C whose kernel is the intersection of kernels of A and B?
Let $V$ be a finite dimensional linear space over a finite field $k$. Let $A$ and $B$ be two endomorphisms of $V$.
Question 1. Is there an endomorphism $C$ of $V$, which is expressed in terms of ...
- 3,897
0
votes
1
answer
263
views
Separability of inner product to a product of Minkowski function and norm
I’ve encountered the following assumption:
Let D be a set such that there exists a Minkowski function $f(u)$ on $\mathbb{R}^l$ and norm $g(v)$ on $\mathbb{R}^m$ such that
$\forall u\in \mathbb{R}^l, \...
- 1
16
votes
4
answers
3k
views
How many minors I need to check to conclude all minors will vanish ?
Given a $m \times n$ matrix $n>m$, I was trying to check if all its $m \times m$ minor vanish.
I remember hearing that one really does not need to check all possible minors in order to conclude ...
- 1,795
1
vote
2
answers
6k
views
Square root of non-positive definite matrix
Finding square root of matrices using Cholesky decomposition is limited to positive definite matrices. Any other method to find square root of matrix which has some diagonal values approximately zero (...
- 11
3
votes
1
answer
490
views
Bounds on operator 2-norms on partial traces of linearly related operators
Consider an arbitary positive semidefinite operator ρ, acting on ℂA ⊗ ℂB ⊗ ℂC, for A,B,C finite. Also, let P be an orthogonal projector on &#...
- 1,444
56
votes
21
answers
18k
views
Wonderful applications of the Vandermonde determinant
This semester I am assisting my mentor teaching a first-year undergraduate course on linear algebra in Peking University, China. And now we have come to the famous Vandermonde determinant, which has ...
1
vote
2
answers
526
views
Eigenvalues and transpose
Please help me with the following question.
Let $F:\mathbb{R}^{k}\to\mathbb{R}^{k}$ be a continuously differentiable mapping;
$F_{n}(x)$ be $n$-th iteration of $F(x)$, i.e. $F_{1}(x)=F(x)$, $F_{n}(x)...
7
votes
3
answers
3k
views
Symmetric subspace of linear operators
This is a question that stemmed from fooling around with unitary t-designs.
Let
\begin{equation}
\mathbb{V} = \mathrm{span} \{\; U^{\otimes t}\; |\; U \in \mathrm{U}(d)\}
\end{equation}
Where $\...
0
votes
1
answer
6k
views
Finding the determinant of a matrix with LU composition
Hi Mathoverflow
I hope you bear with me that my linear algebra knowledge is a little rusty, but I have a question that might potentially very easy to answer. Nevertheless it's been bugging me for a ...
- 105
12
votes
5
answers
2k
views
Is this formulation of the Singular Value Decomposition standard?
In customary formulations of the Singular Value Decomposition or SVD that I have seen,
(e.g., Wikipedia or Gil Strang's textbooks) it is always stated in terms of writing an
$m \times n$ matrix $M$ (...
- 15.3k
16
votes
1
answer
2k
views
Commuting Matrices and the Weak Nullstellensatz
In the Wikipedia article on Hilbert's Nullstensatz,
http://en.wikipedia.org/wiki/Hilbert%27s_Nullstellensatz
the following application of the Weak Nullstensatz is mentioned:
Commuting matrices
...
- 839
4
votes
2
answers
607
views
Invertible elements in monoid rings of unital monoids without non-trivial invertible elements
This question is somewhat related to Tilmans notorious problem in this post. Let $(M,\cdot)$ be a monoid with unit $1$ and set
$$(M,\cdot)^{\times} := \lbrace x \in M \mid \exists y \in M : xy=yx=1 \...
- 25.5k
0
votes
3
answers
498
views
Morphisms between representations
I am looking at the automorphism group $G$ of a graph, represented as permutation matrices. The point in a proof I am trying to understand goes something like this:
"For any permutation matrix $P$ ...
- 1,327
10
votes
3
answers
3k
views
The largest eigenvalue of a "hyperbolic" matrix
Given an integer $n\ge 1$, what is the largest eigenvalue $\lambda_n$ of the matrix $M_n=(m_{ij})_{1\le i,j\le n}$ with the elements $m_{ij}$ equal to $0$ or $1$ according to whether $ij>n$ or $ij\...
- 23k
5
votes
1
answer
2k
views
annihilator/common left multiple of matrix polynomials
Let $A_{n,d}$ be the space of polynomials of degree $d$ or less whose coefficients are real $n\times n$ matrices --- or, if you prefer, the space of matrices whose entries are degree-$d$ polynomials. ...
- 20.2k
2
votes
0
answers
104
views
Noisy bases for linear functions
For any $x \in \mathbb{R}^n$, the following statement is trivially true:
There exists a set $I \subset \mathbb{R}^n$ with $|I| \leq n$ such that for any $x' \in \mathbb{R}^n$, if $x \cdot y = x' \...
- 794
8
votes
3
answers
5k
views
Exact computation of the null-space basis of an integer matrix
Hi all,
Let $\mathbf{A} \in \mathbb{Z}^{M \times N}$. Suppose that $\mathbf{A} \cdot \vec{x} = \vec{0}$, where $\vec{x} \in \mathbb{N}^{N \times 1}$. Does anyone know about a C/C++/Java program that ...
- 81
5
votes
2
answers
917
views
Is the inclusion of Lebesgue spaces compact?
[Disclaimer: this may be a very trivial question; it certainly looks like it ought to have been studied and understood. I started thinking about it this morning when writing some notes for Rellich-...
- 39k
17
votes
1
answer
4k
views
How complicated is infinite-dimensional "undergraduate linear algebra"?
The name "undergraduate linear algebra" in the title is a bit of a joke, and so I don't know how widely spread it is. To wit:
High school linear algebra is the theory of a finite-dimensional vector ...
- 54.6k
11
votes
2
answers
1k
views
An algorithm to find non-trivial linear dependencies
This question is inspired by another MO question about special stratifications of equivariant Grassmannians, that turned out to be a problem of computing non-trivial circuits in a vector matroid. To ...
- 56.6k
14
votes
2
answers
7k
views
What is the dual concept to "annihilator" called, and do any linear algebra textbooks discuss this concept first?
When introducing dual spaces for the first time, most linear algebra textbooks proceed in what seems to me a rather backwards fashion: the annihilator $\{f\in V^*: f(u)=0\quad \forall u\in U\}$ of a ...
4
votes
2
answers
520
views
Can a commutative, associative "multiplication" on an infinite-dimensional vector space be an isomorphism?
Let $V$ be a vector space (over $\mathbb C$, but I don't think it matters), and $m: V\otimes V \to V$ a "multiplication" that is associative and commutative (but I do not demand that it is unital). ...
- 54.6k
6
votes
1
answer
301
views
Orbits in commutative groups.
Let A be finite commutative group say $(Z_m)^h$. I will say that $S \subset A$ is an orbit if exist group $H$
which acts on A such that $S$ is an orbit of $H$.
Can one give a simple characterization ...
- 2,179
1
vote
2
answers
1k
views
Inequality-constrained linear-regression, what is the covariance of the estimator?
If you do a linear regression: $||Ax - e ||^2$, where e is iid Gaussian, mean 0 and variance 1, then your answer is $x_{hat} = (A' A)^{-1} (A' * e)$ and the covariance of $x_{hat}$ is $(A' A)^{-1}$
...
2
votes
1
answer
385
views
Does knowing a conjugation of A to A^T determine eigenvalues of A?
Everybody knows that a square matrix $A$ has the same eigenvalues as
$A^T$. And it is clear that if $A^T=BAB^{-1}$ then $B$ maps eigenvectors
of $A$ to those of $A^T$. But I have not found any ...
- 798
6
votes
4
answers
7k
views
Why do we want to have orthogonal bases in decompositions?
In the decompositions I encountered so far, we all had orthogonal set of bases. For example in Singular Value Decomposition, we had orthogonal singular right and left vectors, in [discrete] cosine ...
- 497
7
votes
1
answer
3k
views
How to resolve a wedge product of vector bundles
Let $X$ be an algebraic variety. Consider an exact sequence
$$0\to A\to B\to C\to 0$$
of vector bundles on $X$. I have seen in different papers the following type resolution of wedge product of $C$ (...
- 2,444
2
votes
0
answers
328
views
What is this decomposition called?
Let $M$ be a positive semi-definite matrix, symmetric with real entries. Then $M$ can be written as $X X^T$. One way is by a Cholesky decomposition (unique for positive definite but not necessarily ...
- 173
6
votes
3
answers
1k
views
Pinching and positive definite matrices
A pinching over $M_n({\mathbb C})$ is an endomorphism $T$ where the $(i,j)$-entry of $T(M)$ is given either by $0$ or by $m_{ij}$, depending on the pair $(i,j)$. Let us say that a pinching is ...
- 52.3k
4
votes
3
answers
3k
views
Making MATLAB svd robust to transpose operation
I'm playing with MATLAB's svd function to compute the svd of
[ 1 4 7 10
2 5 8 11
3 6 9 12 ]
When I type [U1, ~, ~] = svd(...
- 497
5
votes
2
answers
1k
views
Is the operator norm always attained on a $\{0,1\}$-vector?
Given an operator $f\colon R^m\to R^n$, can one always find a non-zero vector
$x\in \{ 0,1 \}^m$ such that $\|f(x)\|/\|x\|\ge0.01\|f\|$? (Here I denote by
$\|\cdot\|$ both the Euclidean norms in $R^m$ ...
- 23k
2
votes
2
answers
369
views
vectors with entries from a finite ring
I've been working recently with vectors over finite fields, but I was hoping to work in a more general setting and consider vectors over finite commutative rings. The question I had is as follows: if ...
- 21
10
votes
7
answers
2k
views
Proof that bases etc. exist in early linear algebra course?
I'm currently struggling to teach a 2nd course on linear algebra (in the UK, not at an Oxbridge quality university: the students have done a 1st course which concentrated upon algorithms you can apply ...
18
votes
3
answers
2k
views
Torsion in GL_n(Z)
Fix some $n \geq 3$. It's hopeless to classify the torsion elements in $\text{GL}_n(\mathbb{Z})$, but I have a couple of less ambitious questions. It's well-known that for any odd prime $p$, the map ...
- 44.8k
5
votes
2
answers
612
views
A Boolean function that is not constant on affine subspaces of large enough dimension
I'm interested in an explicit Boolean function $f \colon \{0,1\}^n \rightarrow \{0,1\}$ with the following property: if $f$ is constant on some affine subspace of $\{0,1\}^n$, then the dimension of ...
