All Questions
Tagged with linear-algebra reference-request
318 questions
3
votes
1
answer
421
views
Inequality for $AB + BA$ when $A,B\geq0$, reference request
Let $A,B\geq0$ be positive semidefinite matrices of arbitrary size $n\times n$. Denote by $\alpha$ and $\beta$ their largest eigenvalues.
It is well-known that the eigenvalues of the expression $AB +...
- 219
1
vote
1
answer
100
views
category of non-welldefined linear maps
I was wondering whether the following category already has been used somewhere and whether it already has been named.
Let us fix a field $k$ (or more generally a ring). An object is just a $k$-vector ...
- 11.1k
1
vote
0
answers
104
views
Convergence rate of Toda/Morse flow
Let $A(t), A_0$ be a $n\times n$ hermitian complex matrices and consider the following matrix flow
\begin{align}
\frac{dA}{dt} &= \left [ C\circ A , A \right ] \\
A(0) &= A_0 \ .
\end{align}
...
- 516
5
votes
1
answer
213
views
Matrix-valued periodic Fibonacci polynomials
Consider the Fibonacci polynomials $f_n(x)$, defined by the recursion $f_n(x)=xf_{n-1}(x)-f_{n-2}(x)$ with initial values $f_0(x)=0$ and $f_1(x)=1$. It is well known that the values of these ...
- 5,622
13
votes
3
answers
2k
views
Linear algebra underlying quantum entanglement?
Hope this question is appropriate. I think I saw certain claims that quantum entanglement is a certain phenomena that can be explained (or modelled) in terms of tensor products in linear algebra. I ...
- 14.3k
7
votes
1
answer
299
views
Lipschitz-continuity of convex polytopes under the Hausdorff metric
Recently, I proved the following Lipschitz-continuity like result for convex polytopes:
Let $A\in\mathbb R^{m\times n}$ and $b,b'\in\mathbb R^m$ be given such that $\{x\,:\,Ax\leq 0\}=\{0\}$ (which ...
5
votes
1
answer
274
views
Precise probability that $m$ random vectors in $n$ dimensional space are nearly-orthogonal
Consider $m$ vectors $v_1,\dots,v_m$ in $\mathbb R^n$, drawn uniformly and independetly from unit sphere. It is pretty straightforward from Chebyshev inequality that
$$
\mathrm P (\forall i\ne j \ |...
- 331
4
votes
0
answers
247
views
Eigenvalues of structured matrices
Let $A=(a_{i,j})$ be an $n\times n$ matrix with $a_{j,j+1}>0,\; 1\leq j\leq n-1,$ and $a_{j,j-2}>0,\; 3\leq j\leq n$, the rest of the entries are zeros.
Is the following fact known:
All ...
- 91.8k
44
votes
2
answers
2k
views
Fermat's Last Theorem for integer matrices
Some years ago I was asked by a friend if Fermat's Last Theorem was true for matrices. It is pretty easy to convince oneself that it is not the case, and in fact the following statement occurs ...
- 1,889
4
votes
1
answer
511
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Invariants of symmetric forms with respect to the symplectic group
Take a 6-dimensional vector space $V$ (for simplicity, over $\mathbb{C}$) and play the following game (for example, by employing the online Lie program): consider the 21-dimensional space $S^2V^*$ of ...
- 2,527
8
votes
3
answers
663
views
Representation theorem for matrices (reference request)
Motivation. If $A \in \mathbb{C}^{n \times n}$ is self-adjoint (or, more generally, normal), then we all know that
$$
A = \sum_{k=1}^n \lambda_k \, h_k \otimes h_k,
$$
where $\lambda_1,\dots,\lambda_n$...
- 12.6k
3
votes
1
answer
3k
views
Singular value decomposition of random rectangular matrices
Let $A$ be a $m\times n$ real matrix, whose entries are independent, identically distributed random variables, following standard normal distributions (mean zero and unit variance).
What is the ...
- 884
0
votes
0
answers
94
views
Neat expresion for an anti-symmetric matrix
Fix a column vector $\pmb{v}$ and consider the cross product $\pmb{v}^T\times\pmb{x}^T$ for any column vector $\pmb{x}\in\mathbb{R}^3$. One can write
$$\pmb{v}^T\times\pmb{x}^T=A(\pmb{v})\pmb{x}$$
for ...
- 43.2k
1
vote
0
answers
126
views
Algebraic structures on graphs
There are many algebraic structures linked to graphs.
For example one can find zero divisor graphs $[1]$, $[2]$ and many other graphs.
Does there exist any survey paper which characterizes all the ...
- 444
2
votes
0
answers
1k
views
Applications of linear algebra in the design of aircraft [closed]
David Lay mentioned one application of linear algebra in the design of aircraft in the introductory part of chapter 2 of his book:
[...] A computer creates a model of the surface by first ...
- 101
0
votes
0
answers
84
views
Relation between two matrices associated with a positive definite function
Let $f:\mathbb{R}^N \to \mathbb{R}$ be a positive definite function. Let $$g(h) = \int_{\mathbb{R}^N}f(x)f(h-x)\mathrm{d}x$$ Due to Bochner's and Parseval's theorems, $g$ is also a positive definite ...
- 698
10
votes
1
answer
483
views
functors $\text{Vect} \to \text{Vect}$ that preserve filtered and sifted colimits
I'm considering various functors from the category $\text{Vect}$ of real vector spaces to itself, and would like to know that they preserve filtered colimits and possibly even sifted colimits. The ...
- 1,092
4
votes
0
answers
144
views
A Pythagorian inequality characterization of inner-product spaces
Let $(X,\|\cdot\|)$ be a real normed space. For any points $A$ and $B$ in $X$, let $AB:=\|A-B\|$. Suppose that for any points $A$ and $B$ in $X$ and any straight line $\ell\subseteq X$ such that $B\...
- 128k
10
votes
1
answer
262
views
What is the hidden symmetry behind four generic planes in $\mathbb{R}^4$?
Consider the action of $\operatorname{GL}(\mathbb{R}^4)$ on the Grassmannian of 2-dimensional subspaces of $\mathbb{R}^4$. In experiments, I observe that four randomly drawn points in this space are ...
- 7,633
6
votes
1
answer
237
views
Determinantal questions on Alternate Sign Matrices
Let $\mathcal{A}_n$ be the set of all Alternating Sign Matrices (ASM) of size $n\times n$. The cardinality $\#\mathcal{A}_n$ is well-known
$$\#\mathcal{A}_n=\prod_{k=0}^{n-1} \frac{(3k+1)!}{(n+k)!}.$$
...
- 43.2k
2
votes
0
answers
148
views
Generalization of Farkas' Lemma to Hermitian Matrices
I recently stumbled upon a well-known version of Farkas' Lemma which, roughly speaking, I would like to generalize from real vectors to hermitian matrices, as it seems promising for something else I ...
3
votes
0
answers
55
views
What is the precise definition of a quadratic form of Minkowski type (in the infinite case)?
I've been trying to understand a construction in the paper "Degree Growth of Meromorphic Surface Maps" by Bouksom, Favre and Jonsson. In it they state,
In fact, the completion can be characterized ...
- 31
3
votes
0
answers
180
views
Automorphisms of infinite matrix algebra
This is a similar question to one that I posted in MSE a few days ago.
I recently came across this paper from Alahmedi, Alsulami, Jain and Zelmanov, which quoted the following result for $M_\infty(K)$...
- 195
3
votes
1
answer
777
views
Lower bound of the expectation of the product of inner products of random vectors
I encountered the following value in my research:
Let $n,m$ be some integer. Suppose $\alpha_1,\dots,\alpha_m$ are unit vectors in $\mathbb{R}^n$.
Denote
$$
L = \mathop{\mathrm{E}}_x[ \prod_{1\...
- 1,551
3
votes
0
answers
360
views
Do we know what the impulse to "introduce" the Jordan canonical form was?
Mo-ers,
Do you know how it was that the study of the Jordan canonical form began?
There are certain things that may be said once one has thought about the matter: for instance, one can say that the ...
- 87
3
votes
0
answers
122
views
Algebra of block matrices with scalar diagonals
I am interested in block matrices $A$, that is $A\in M_{n\times n}(R)$ where $R=M_{s\times s}(k)$ and $k$ is a field, such that for every positive integer $m$ the matrix $A^m$ has only scalar blocks ...
- 3,041
4
votes
1
answer
237
views
Convex Hull of Outer Products of (Normalised) Nonnegative Vectors
If I define $\mathcal{A} = \{ xx^T : x \in \mathbb{R}^d, \| x \|_2 \leqslant 1 \}$, then (assuming I recall correctly) it is known that the convex hull of $\mathcal{A}$ is given by
\begin{align}
\...
- 801
5
votes
0
answers
150
views
monomer-dimer tiling of a Young diagram
As a modest start, I propose the below problem for a special set of partitions. Perhaps it is known.
Let $\lambda_n=(n,n-1,\dots,2,1)$ be the staircase partition and its corresponding Young diagram $...
- 43.2k
5
votes
1
answer
515
views
Do matrices with only elements along the main and anti-diagonals have a name?
To expand upon the title, I am wondering if there is a specific name for square matrices of the form: $$M = \begin{bmatrix} a_{11} & 0 & \cdots & 0 & \cdots & & 0 & b_{1n} ...
- 169
3
votes
1
answer
486
views
Is this line of thought (using linear algebra to get number theoretic results) already being pursued in the literature?
Let $Log(n) = \sum_{i=1}^r \alpha_i \cdot e_i$, where $n = \prod_{i=1}^r p_i^{\alpha_i}$ and $p_i$ is the $i$-th prime, $\alpha_i \ge 0$, $e_i$ is the $i$-th standard basis vector. For example $6 = 2\...
user6671
6
votes
0
answers
210
views
Lambek calculus, linear logic, and linear algebra
In his 1958 paper, The Mathematics of Sentence Structure, Joachim Lambek introduced the Lambek calculus. In modern terms, it could be understood as a syntax for biclosed
monoidal categories, and he ...
- 9,201
1
vote
0
answers
159
views
Non-trivial ways for generating matrices $A$ for which $A + A^T$ is positive-definite?
Disclaimer: This might be an SE question, but I'm not quite sure...
Thanks in advance!
Setup
So, it is known (see Proposition 5.2) that if $A + A^T$ is positive-definite then $A$ must be a $P$-...
- 6,853
1
vote
1
answer
151
views
Counting monomials in skew-symmetric+diagonal matrices
This question is motivated by Richard Stanley's answer to this MO question.
Let $g(n)$ be the number of distinct monomials in the expansion of the determinant of an $n\times n$ generic "skew-...
- 43.2k
3
votes
0
answers
114
views
Jacobian of the action of a matrix on a Grassmannian
I'm looking for a reference concerning a calculation found in Furstenberg's 1963 paper "Non-commuting random products".
Lemma 8.8 of this paper states that if one takes a $d\times d$ invertible real ...
- 4,304
18
votes
2
answers
1k
views
Is a matrix similar to its transpose over $\mathbb{Z}_p$?
Is every $n \times n$ matrix with entries in $\mathbb{Z}_p$ (or even $\mathbb{Z}$) conjugate to its transpose via a matrix in $GL_n(\mathbb{Z}_p)$?
On the one hand, I know the analogous fact is false ...
- 2,242
1
vote
1
answer
95
views
A question on a special "metric"
Suppose we have a function $F: [a,b]^n \to \mathcal{M}_{n \times n }(\mathbb{R})$ where $\mathcal{M}_{n \times n }(\mathbb{R})$ is the space of $n \times n$ real matrices, a compact set $B \subset \...
- 113
17
votes
2
answers
2k
views
The Lefschetz operator
Let $\omega=\sum_{i=1}^n dx_i\wedge dy_i\in\bigwedge^2(\mathbb{R}^{2n})^*$ be a standard symplectic form. The following result is due to Lefschetz:
For $k\leq n$, the Lefschetz operator
$L^{n-k}:\...
- 28k
4
votes
2
answers
1k
views
Reference request: Oldest linear algebra books with exercises?
Inspired by the recent success of my "soft question" here, I also have to ask, what are some of the oldest linear algebra books out there with exercises? I'm fine with or without solutions, either way....
5
votes
2
answers
250
views
Eigenvalue density of a symmetric tridiagonal matrix
Let $A_n\in\mathbb{R}^{n\times n}$ be defined as
$$
A_n=\begin{bmatrix} a & b & 0 & \cdots & \cdots & 0 & 0\\ b & a & b & \cdots & \cdots & 0 & 0\\ 0 &...
- 2,712
2
votes
1
answer
553
views
Upper Bounds on the Largest Eigenvalue of Jacobi Matrices
Suppose I have a symmetric tridiagonal (Jacobi) matrix in the following form:
$ \begin{pmatrix}
1 & a_{1} & 0 & ... & 0 \\\
a_{1} & 1 & a_{2} & & ... \\\
0 & a_{...
- 137
3
votes
1
answer
325
views
Reference on completely positive maps which are isometries
Let $\Phi:\mathcal{L}(H)\rightarrow \mathcal{L}(K)$ be a completely positive map sending positive self-adoint operators on a finite-dimensional Hilbert space $H$ to positive self-adoint operators on a ...
- 627
4
votes
3
answers
382
views
Extending a continuous map over projective space
Let $X = P^{n-1}(\Bbb C)$ ($(n-1)$-dimensional projective space) with $n \geq 3$, and let $K \subset X$ denote a compact subset. I have a bijective, continuous map $\phi:K \to K$ which satisfies the ...
- 356
4
votes
0
answers
98
views
Ref. request: Enumerating elements of Bruhat cells
Given a field $F$ and a natural number $n$, let $B$ be the group of lower triangular, invertible $n \times n$ matrices over $F$. Then
$$GL_n(F) = \biguplus_{\pi \in S_n} B \pi B,$$
where we embed the ...
- 809
34
votes
4
answers
2k
views
If $A,B$ are upper triangular matrices such that $AX=XA\implies BX=XB$ for upper triangular $X$, is $B$ a polynomial in $A$?
A professor of mine told me that this is true, but he doesn't remember what the proof was or where to find it, and I haven't been able to find a source for it yet. As such I am looking for one here.
...
- 343
5
votes
1
answer
369
views
Connection between Gram matrix and Riemannian invariants?
Recall that the Gram matrix of vectors $v_1, \dots, v_k\in\mathbb{R}^n$ is the $k\times k$ matrix $G_{ij}=(v_i,v_j)$. Now suppose that the vectors $v_i$ have been sampled uniformly from some ...
- 505
1
vote
0
answers
620
views
On the basis of a finite dimensional vector space (revised)
Revision in response to the comments to earlier version:
To introduce the notion of a basis of a finite dimensional vector space over an arbitrary field $\Lambda$, without performing any computation ...
- 51
2
votes
1
answer
255
views
Efficient algorithm for solving a convex quadratic program [duplicate]
Let $A \in \mathbb{R}^{n \times m}$ and $b \in \mathbb{R}^n$. Suppose $m \ll n$. How to solve this quadratic program efficiently?
$$\min_{x \in \mathbb{R}^n} \frac{1}{2} x^\top AA^\top x + b^\top x$$
- 422
11
votes
1
answer
731
views
Reference request: Volume 2 of Abhyankar's lectures on algebra?
Abhyankar has a magnificent, if meandering (check them out if you want to see what I mean), set of lectures on algebra.
The description:
This book is a timely survey of much of the algebra developed ...
- 111
1
vote
0
answers
148
views
Traces in associative algebras
Are there some books or papers about the general definition of traces:
If $\mathscr{A}$ is an associative algebra over $K$ then the space of traces is the set of all linear functionals $\tau:\mathscr{...
- 283
2
votes
0
answers
55
views
Lower bounds on eigenvalues of Lyapunov solutions
Let $A\in\mathbb{R}^{n\times n}$, $B\in\mathbb{R}^{n\times m}$ and let $X\in\mathbb{R}^{n\times n}$, $X=X^\top>0$ be the solution of the following Lyapunov algebraic equation
$$
AX+XA^\top=-BB^\top....
- 2,712