All Questions
Tagged with linear-algebra reference-request
318 questions
4
votes
1
answer
313
views
The characteristic polynomial of the product of two linear recurrences
Let $\mathbb{F}$ be a field and let $(a_n)_{n \geq 0}$, $(b_n)_{n \geq 0}$ be two linear recurrences with terms in $\mathbb{F}$ and respective characteristic polynomials $f(X), g(X) \in \mathbb{F}[X]$....
user40023
4
votes
1
answer
630
views
Can sparse matrices satisfy the Null Space Property?
Definition A matrix $A \in \mathbb{C}^{m \times N}$ with $m < N$ satisfies the Null Space Property (NSP) of order $s$ if
$$\|x_S\|_1 < \|x_{\bar{S}}\|_1, \quad \forall x \in \ker A \setminus \{...
- 837
4
votes
1
answer
408
views
Proof of Levinson-Durbin algorithm
Is there any article or reference book with a full proof of the Levinson-Durbin algorithm used for solving linear system with a Toeplitz matrix ?
- 41
4
votes
1
answer
214
views
The d-dimensional matrix with columns (1,0,0…), (1/2,1/2,0,…), (1/3,1/3,1/3,0,…),…, (1/d,1/d,…,1/d)
During the course of physics research on nonequilibirum statistical mechanics involving the theory of majorization, I have come across a linear transformation on a d-dimensional vector space that I ...
- 41
4
votes
0
answers
284
views
Institutional approach to linear algebra
In Diaconescu's book Institution Independent Model Theory, it is mentioned on p. 37 that linear algebra can be viewed as an institution. Specifically, we have the following
Definition. An institution ...
- 10.1k
4
votes
0
answers
181
views
Fuss-Catalan: how does equality of these determinants hold?
There are many ways that the Catalan numbers seemed to have been generalized, one among them is through what Graham-Knuth-Patashnik (in Concrete Mathematics) dubbed as the Fuss-Catalan numbers
$\frac1{...
- 43.2k
4
votes
0
answers
112
views
Duality for finite quotient groups of finitely generated free abelian groups
$\newcommand{\Z}{{\Bbb Z}}
\newcommand{\Q}{{\Bbb Q}}
\newcommand{\Hom}{{\rm Hom}}
$ The following lemma is certainly known.
Lemma (well-known).
Let $B$ be a lattice (that is, a finitely generated ...
- 14.2k
4
votes
0
answers
1k
views
Reference for matrices with all eigenvalues 1 or -1
In a homological algebra problem I am in the situation that I have an invertible (over $\mathbb{Z}$) integer matrix $X$ and a permutation matrix $Y$ such that $N:=XY$ is a matrix with all eigenvalues ...
- 26.5k
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
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
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,$$
where we embed the ...
- 809
4
votes
0
answers
245
views
On the sum of the first row of the inverse of a certain symmetric Toeplitz matrix
(i) Consider a Toeplitz matrix $A_n = (a_{i, j})_{1 \le i, j \le n}$ of size $n$ defined as follows:
$$ a_{i, j} := |i-j|^{-1/2}, \text{ if } i \ne j; \ \ a_{i, j} := 2, \text{ if }i = j. $$
Let $...
- 195
4
votes
1
answer
234
views
A trivialization of an almost complex structure
Recently, I have been studying the Carleman Similiarity Principle, which is used to study the regularity and unique continuation of J-holomorphic curves.
Roughly, one takes a solution $ u $ of a ...
- 337
4
votes
0
answers
136
views
What do we know about the generalized eigenvalue problem involving a projector?
Consider a matrix $A\in\mathbb{R}^{n\times n}$ and a projector $P\in\mathbb{R}^{n\times n}$.
Are there results regarding the generalized eigenpairs $(v,\lambda)$ of the generalized eigenproblem
$$...
- 325
4
votes
0
answers
167
views
Reduction of size of orthogonal matrices.
While experimenting with orthogonal vectors I've noticed the
following transformation: If
$$
A = \begin{bmatrix}z & r \cr
c & B\end{bmatrix}
$$
is orthogonal, $z$ ...
- 798
4
votes
0
answers
189
views
Slices of Simplices that are Simplices, Reference?
I am trying to find a reference for the following fact. It is elementary and not hard to prove, but I haven't been able to find the question treated anywhere.
Let $A$ be an $l\times n$ matrix with ...
- 41
4
votes
0
answers
352
views
"Cholesky decomposition" X=YY* for p-adic matrices?
Let $E/F$ be a quadratic extension of $p$-adic fields. Consider $M_n(E)$ with the unitary (aka 2nd kind) involution $X \mapsto \sigma(X)^{tr}$, where $\sigma(X)$ denotes the entry-wise application of $...
- 932
3
votes
4
answers
570
views
A polynomial homomorphism from Gl to the group of units is a power of the determinant
I was browsing MO and stumbled upon this post, and I got very curious. I searched for about half an hour and could not find a proof for the statement that any polynomial group homomorphism $\mathrm{Gl}...
- 4,902
3
votes
3
answers
1k
views
A table for irreducible integral representation of finite cyclic groups
Is there such a table where the irreducible integral representations of finite cyclic groups
are listed?
Edited:
Thanks for Todd Leason's comment.Acutally,i want to know all inequivalent ...
- 593
3
votes
3
answers
1k
views
Applications of rank factorization or full rank decomposition [closed]
I am teaching a course on linear algebra and came to this theorem: every $m \times n$ matrix $A$ with rank $r$ admits a factorization $A = CR$ where $C$ is an $m \times r$ matrix and $R$ is an $r \...
- 357
3
votes
1
answer
836
views
Solving multilinear equations
Let $N=\{1,2,\ldots,n\}$. Suppose we are given $n$ equations, with each equation taking the form $\sum_{A\subseteq N}\left(c_A \prod_{i\in A}x_i \right) = 0$, where each $c_A$ is a real number ...
- 239
3
votes
2
answers
939
views
Positive definiteness of infinite tridiagonal matrices
I am interested in the following problem: I have an infinite symmetric tridiagonal matrix
$$
A=
\begin{bmatrix}
a_1 & b_1 & & & \\
b_1 & a_2 & b_2 & & \...
- 692
3
votes
1
answer
156
views
How can I show $\{\mathbf{x}: \dim (\ker M_1(\mathbf{x}) \cap \ker M_2(\mathbf{x})) \geq C \}$ is an affine variety?
Let $M_1(\mathbf{x})$ and $M_2(\mathbf{x})$ be $m$ by $m$ matrices with each entry a homogeneous form in $\mathbb{C}[x_1, \ldots, x_n]$.
I would like to show that
$$
\{ \mathbf{x} \in \mathbb{A}^n_{\...
- 3,625
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
3
votes
2
answers
976
views
Positive solutions of linear systems with a diagonally dominant matrix
Given a real linear system ($\mathbf{A}\mathbf{x} = \mathbf{b}$), is there any result regarding the positiveness of the solution $\mathbf{x}^*$ considering that $\mathbf{A}$ is diagonally dominant? (...
- 342
3
votes
1
answer
166
views
The spectral radius of a modified graph
Let $H$ be a graph and let $G=H \vee K_{1}$ be obtained by creating a new vertex and joining it to every vertex in $H$.
This situation has many different names: $G$ is called the cone or the ...
- 6,962
3
votes
1
answer
5k
views
Relation between the eigenvalues of a block matrix and the eigenvalues of its diagonal blocks
Consider the $(m+n) \times (m+n)$ block matrix
$$M = \begin{bmatrix} A & B \\ C & D \end{bmatrix}$$
I need references where they are talking about the relation between the eigenvalues of $M$ ...
- 1,269
3
votes
1
answer
655
views
Upper bounds on the condition number of the eigenvector matrix
Let $A$ be an $n\times n$ real matrix with entries in a fixed interval $[a_\min,a_\max]$, with $a_\min$, $a_\max>0$.
Question: Are there any upper bounds on the condition number of the ...
- 2,712
3
votes
1
answer
389
views
Galois deformations with Panchiskin condition
Let $L/\mathbf{Q}_p$ be a finite extension and we consider a fixed $L$-linear representation $V$ of the absolute Galois group $G:=\operatorname{Gal}(\overline{\mathbf{Q}}_p/\mathbf{Q}_p)$. Assume that ...
- 63
3
votes
1
answer
3k
views
Diagonalize the simultaneous matrices and its background [closed]
For two $n \times n$ nonnegative definite Hermitian matrices $A$ and $B$ over the real number field $\mathbb R$:
Question1:Is there always a
nonsingular matrix $P$ over the same
field $F$ which ...
- 8,071
3
votes
1
answer
327
views
Derivative norm estimates
Assume $\Phi$ is some diffeomorphism of a certain manifold. Let $\Phi^{-1}$ denote the inverse map and let $(D\Phi)^{-1}$ denote the matrix inverse of $D\Phi$.
QUESTION. Does this norm estimate hold? ...
- 43.2k
3
votes
1
answer
165
views
The inverse of a symbolic matrix (with reciprocal binomials) has Laurent entries
Recalling the $q$-binomials (Gaussian polynomials). Let $[n]_q!=\prod_{j=1}^n\frac{1-q^j}{1-q}$ and
$\binom{n}k_q=\frac{[n]_q!}{[k]_q!\cdot[n-k]_q!}$.
Now, consider the $n\times n$ matrix $\mathbf{M}...
- 43.2k
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
3
votes
1
answer
779
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
1
answer
597
views
Has anybody seen my missing lemma?
I think I have a proof of the following elementary lemma (although I only need the case in which the two flags are "in general position", i.e., $F^d \cap G^i$ is minimal given the dimensions of the ...
- 7,338
3
votes
1
answer
270
views
What is the name of this measure of matrix "degenerateness"
Given a spanning set, consider the minimum number of vectors that you must remove in order to make it no longer span. What is this number called?
If the vectors are columns in a matrix $\Phi$, then ...
- 7,633
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
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
3
votes
1
answer
232
views
Non-degeneracy in hyperplane intersections of canonical curves
Let $C$ be a smooth projective non-hyperelliptic curve over $\mathbb{C}$ of genus $g = 4$. The canonical bundle $\omega_C$ induces a canonical embedding $C \longrightarrow \mathbb{CP}^3 $ such that $C$...
- 343
3
votes
1
answer
203
views
Results on Boolean matrices
Matrices with entries in the finite field of two elements $\mathbb{F}_2$, and with the usual operations of matrix addition and multiplication, have been intensively studied, especially due to their ...
- 31
3
votes
1
answer
175
views
Is there a classical textbook/reference on numerical discretization schemes?
I found that it is relatively easy to find a book that discusses Euler discretization or Runge-Kutta discretization, but I am not aware of one that is well-known and/or common knowledge (i.e., field-...
- 479
3
votes
1
answer
552
views
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:
$$
\...
3
votes
1
answer
767
views
Linear algebra of finite abelian groups
If $f: V \to W$ is a surjective homomorphism of vector spaces, and we have fixed a basis for $V$, it is always possible to find a basis for $W$ such that the matrix associated to $\phi$ in the two ...
- 133
3
votes
1
answer
621
views
Largest eigenvalue of a periodic Jacobi matrix
There is a vast literature on Jacobi matrices, I just don't know where to start looking. I'm interested in estimating the largest eigenvalue of the $n\times n$ periodic Jacobi matrix $D+P+P^{-1}$, ...
- 11.1k
3
votes
0
answers
198
views
On a paper of von Neumann
Let $H$ be a Hilbert space and $T: H \to H$ be a contraction. In Eine Spektraltheorie für allgemeine Operatoren eines unitären Raumes, von Neumann proved the inequality
$$
\lVert p(T)\rVert \leq \sup \...
- 31
3
votes
0
answers
235
views
Do you know this formula for the scalar product in barycentric coordinates?
I've found a formula for a scalar product in barycentric coordinates which I think is pretty cool. I hope that it's new. Is it?
Suppose that you have points $x_1,\dots,x_n$ sitting in general position ...
- 1,058
3
votes
0
answers
185
views
"Circulant-Vandermonde" matrix: in search of a formula
An $n\times n$ circulant matrix $\mathbf{X}_n$ has the form
\begin{align}
\mathbf{X}_n= \begin{bmatrix}
x_1 & x_2 & \cdots & x_{n-1} & x_n \\
x_2 & x_3 & \cdots & x_n&...
- 43.2k
3
votes
0
answers
207
views
On a variation of the Vandermonde matrix
The ubiquitous Vandermonde matrix, of entries $(x_i^{j-1})_{i,j}^{1,n}$, and its determinant
$$\prod_{i<j}^{1,n}(x_j-x_i)$$
have found many utilities in Combinatorics and Physics, among other ...
- 43.2k
3
votes
0
answers
66
views
Classification or bounds on "totally symmetric" arrangements of subspaces
Let $F$ be a field and let $W_1, \dots, W_k \subset F^n$ be a collection of $d$-dimensional subspaces of $F^n$ such that $W_i \cap W_j = \{0\}$ for all indices $i,j$. We say that such an arrangement ...
- 2,830
3
votes
0
answers
147
views
Convolution integral and its matrix representation
My background is chemistry and I was exploring some one dimensional deconvolution problems i.e., resolution of two or more overlapping peaks. A lot of excellent work was done in the 1970-80s. However, ...
- 879