All Questions
Tagged with linear-algebra reference-request
318 questions
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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 ...
4
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1
answer
237
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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}
\...
- 6,351
5
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4
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839
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Norm bounds on spectral variation and eigenvalue variation
Let $A$ and $B$ be two matrices of eigenvalues $\lambda_i$ and $\mu_i$, respectively.
The spectral variation of $B$ w.r.t. $A$ and the eigenvalue variation of $B$ and $A$ are, respectively,
\begin{...
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5
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0
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150
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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
3
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1
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486
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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\...
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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
10
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1
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537
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Coefficient-wise powers of matrices. Reference wanted
Let $K$ be a commutative field and ${\rm M}_n (K)$ be the ring of $n\times n$ square matrices with coefficients in $K$ ($n\geqslant 1$ is an integer). For $k\geqslant 1$ and $A =(a_{ij})_{1\leqslant i,...
- 66.3k
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0
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159
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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 $P$-...
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1
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1
answer
151
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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
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0
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114
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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
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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 ...
- 3,813
1
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1
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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 \...
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2
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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....
- 188k
5
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2
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250
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Eigenvalue density of a symmetric tridiagonal matrix
Let $A_n\in\mathbb{R}^{n\times n}$ be defined as
$$
A_n=\begin{bmatrix} a & b & 0 & \cdots & \cdots & 0 & 0\\ b & a & b & \cdots & \cdots & 0 & 0\\ 0 &...
- 24.2k
7
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3
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1k
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Determinant of correlation matrix of autoregressive model
I wonder if there is a paper that can point out how to compute the determinant of a $d \times d$ autoregressive correlation matrix of the form
$$R = \begin{pmatrix}
1 & r & \cdots & r^{d-...
- 43.2k
2
votes
1
answer
553
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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_{...
- 79.9k
11
votes
1
answer
731
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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 ...
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3
votes
1
answer
325
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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 ...
- 3,984
4
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3
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382
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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 ...
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0
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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 ...
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34
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4
answers
2k
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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.
...
1
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0
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620
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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 ...
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5
votes
1
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369
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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 ...
- 96.4k
2
votes
1
answer
255
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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?
$$\min_{x \in \mathbb{R}^n} \frac{1}{2} x^\top AA^\top x + b^\top x$$
1
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0
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148
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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{...
- 63.9k
0
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1
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150
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General results regarding linear separability?
I'm reading up on the theory behind support vector machines and would like a good reference with some general results about linear separability.
Specifically, questions like below:
Given two ...
2
votes
0
answers
55
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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
$$
AX+XA^\top=-BB^\top....
- 2,712
4
votes
1
answer
414
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A Handbook of Matrix Factorizations
I am looking for a good collection of facts regarding the various types of matrix factorizations, something like a "Handbook of Matrix Factorizations" or a very-thorough review paper. I am hoping for ...
- 96.4k
6
votes
1
answer
257
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Families of subsets whose characteristic vectors are spanning sets
Let $X$ be a finite set and $\mathbb CX$ be a vector space with basis $X$. If $Y\subseteq X$ is a subset, then by the characteristic vector of $Y$ I mean $\sum_{y\in Y}y$.
My question is:
...
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3
votes
1
answer
552
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Is this result on an unconstrained inverse quadratic programming problem new or known already?
Is this problem and solution actually new, or has someone done this earlier?
The details can be found in the preprint: arxiv:1701.01477.
Let us consider a direct quadratic programming problem:
$$
\...
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2
votes
1
answer
217
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Diagonalising a symmetric matrix with polynomial entries
Suppose I have a symmetric $2$ by $2$ matrix $M$ whose $(i,j)$-th entry $F_{i,j}(\mathbf{x})$ belongs to $\mathbb{R}[x_1, \ldots, x_n]$ for each $i,j$. I know that for each $\mathbf{x} \in \mathbb{R}^...
- 22.5k
8
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2
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950
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Best known bounds on (border) ranks of small matrix multiplication tensors?
The $(m,n,p)$-matrix multiplication tensor is a representation of the bilinear map $T\colon\mathbb{R}^{m\times n}\times\mathbb{R}^{n\times p}\rightarrow\mathbb{R}^{m\times p}$ given by $T(A,B)=AB$. ...
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0
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98
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Eigenvalues of a sequence of matrices involving the divisor function
Let $A_{n,k},k=1,\ldots,n$ be a sequence of $n\times n$ upper triangular matrices where $A_{n,1}=I_n$ and $A_{n,k},\quad 2\leq k\leq n$ be a regularly shifted and scaled matrix, with $P_{n,k}$ an $n\...
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11
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1
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How do computer algebra packages like Sagemath implement rank of a matrix
I am not sure if this is the right place to ask this question, but I believe there will be people here who do computations on computer algebra packages like Sage in their work.
I have been using ...
- 20.2k
5
votes
1
answer
315
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Rank-constrained least-squares solution of the Sylvester matrix equation
For the Sylvester matrix equation $AX+XB=C$, I want to find the least-squares solution $X$ via
$$\begin{array}{ll} \text{minimize} & \| AX + XB - C \|_{\text{F}}^2\\ \text{subject to} & \mbox{...
4
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0
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On the sum of the first row of the inverse of a certain symmetric Toeplitz matrix
(i) Consider a Toeplitz matrix $A_n = (a_{i, j})_{1 \le i, j \le n}$ of size $n$ defined as follows:
$$ a_{i, j} := |i-j|^{-1/2}, \text{ if } i \ne j; \ \ a_{i, j} := 2, \text{ if }i = j. $$
Let $...
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0
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283
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A symmetric matrix with nonzero principal minors is cogredient to a diagonal matrix via an upper triangular
A paper I'm reading in representation theory states the following result:
Let $F$ be a field of characteristic zero, and $x$ a symmetric matrix in $M_n(F)$ all of whose principal minors are not zero. ...
- 6,180
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0
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422
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Difference between largest two eigenvalues of a graph Laplacian
The difference between the smallest eigenvalue and the next-smallest of a graph Laplacian (equivalently, the difference between the largest and next-largest of the random walk Markov chain on the ...
- 1,068
2
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0
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275
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Maximum spectral norm of matrices with given anti-Hermitian part and Hermitian part's spectrum
Let $M\in M_n(\mathbb C)$ be a $n\times n$ matrix over the complex field. It can be written uniquely as $M=H+A$, where $H=H^*$ denotes its Hermitian part and $A=-A^*$ its anti-Hermitian part.
Its ...
- 21
2
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2
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446
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Entrywise modulus matrix and the largest eigenvector
Disclaimer. This is a cross-post from math.SE where I asked a variant of this question two days ago which has been positively received but not has not received any answers.
Let $A$ be a complex ...
- 4,361
3
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0
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105
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Can one do better than using general purpose determinant algorithms when using the Fisher-Kasteleyn-Temperley method for perfect matchings?
Questions.
(numerical.generalPfaffian) Is it proved anywhere that in general it is not easier0 to calculate the determinant (over $\mathbb{Q}$) of the skew-symmetric signed adjacency matrix defined ...
- 6,051
2
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0
answers
102
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Eigenvalue distribution for a real-valued random matrix with correlated Gaussian entries
I'm working on an application where I would greatly benefit from knowing the distributions of the eigenvalues of a real-valued random matrix whose elements can be assumed to be Gaussian, but where I ...
- 321
4
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2
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2k
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Advanced reference and roadmap about random matrices theory
There is few posts on MO that asked about reference on this topic, and I found some difficulty during the process of getting myself into the subject so here is the question.
I really want to hear ...
- 321
1
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0
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188
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Maximum singular value of sum of an Hermitian and an anti-Hermitian matrix
Let $H$ be an $n\times n$ Hermitian matrix and $A$ an $n\times n$ anti-Hermitian matrix, i.e. $H^\dagger = H$, $A^\dagger = -A$. Consider their sum $S= H+A$. Let $\{\sigma_i(S)\}_{i=1,\dots,r}$ denote ...
- 11
2
votes
0
answers
29
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Terminology question- Antihermitian elements
Let $V$ be a vector-space over a field $F$, and let $B$ be a non-degenerate bilinear form on $V$.
Question: Is there a common term to call an operator $x\in End(V)$ satisfying $$B(xu,v)+B(u,xv)=0\...
- 993
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0
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43
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Sequence of neighboring orthogonal vectors
Let $v_1v_2...v_n$ be a sequence of vectors in $\mathbb{F}_2^{m}$. We say that this sequence is neighboring orthogonal if $\langle v_i, v_{i+1}\rangle = 0$ for each $i\in \{1,...,n\}$, where for each $...
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2
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209
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Geometrical interpretation of pictures transforms and other "high dimensional everyday objects"
During the preparation of a general audience talk on why mathematicians use dimensions higher than three (or four) even for concrete applications, I came up with the following enjoyable observation : ...
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10
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3
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830
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Find the inverse of a matrix that is very similar to the Hilbert matrix
The standard Hilbert matrix $H$ is given by
$$H_{ij}=\frac{1}{i+j-1},$$
and it has an inverse given for example in this MO question.
Now I have encountered a matrix $M$ of similar form, namely,
$$...
2
votes
1
answer
325
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Determinant and inverse of a "stars and stripes" matrix
This is a variant of another MO question. Consider the matrix
$$M_n:=\begin{bmatrix}c_1& a & b&a& \ddots & a \\ b & c_2 & a& b&\ddots & b\\ a & b & c_3&...
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27
votes
6
answers
6k
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Origin of exact sequences
I have seen exact sequences appearing a lot in algebraic texts with different purposes. But I've never seen names of the people associated with it. Also I don't understand what's so good about showing ...
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