All Questions
Tagged with matrices reference-request
190 questions
2
votes
0
answers
233
views
Roots of determinant of matrix with polynomial entries — a generalization
For $1 \le i, j \le k$, consider $\rho_{ij}$ which are equal to either zero or one such that $\rho_{ii}=1$ and $\rho_{ij}=0$ if and only if $\rho_{ji}=0$. How to find the zeros of the determinant of ...
- 1,269
2
votes
1
answer
417
views
Roots of determinant of matrix with polynomial entries
Let $p_1, p_2,\dots, p_n$ and $q_1,q_2,\dots,q_n$ be a collection of complex polynomials. Let $A$ be a $n \times n$ matrix satisfying
$$a_{ij} = \begin{cases} p_i(x) & \text{ if } i = j, \\ q_i(x)...
- 1,269
11
votes
1
answer
1k
views
A square root inequality for symmetric matrices?
In this post all my matrices will be $\mathbb R^{N\times N}$ symmetric positive semi-definite (psd), but I am also interested in the Hermitian case. In particular the square root $A^{\frac 12}$ of a ...
- 5,380
3
votes
0
answers
39
views
A non-singularity property for sets of real matrices
Let $M_N(\mathbb{R})$ be the ring of $N\times N$ real matrices. We say that a couple $(\mathcal{U},\mathcal{V})$, with $\mathcal{U},\mathcal{V}\subseteq M_N(\mathbb{R})$ is admissible if, for every $A\...
- 943
9
votes
3
answers
589
views
Subgroup of $\mathrm{GL}_n$ stabilizing linear subspace skew-symmetric matrices
I am currently reading "Schiffer variations and the generic Torelli theorem for hypersurfaces" by Voisin, where it is claimed that the subgroup of $\mathrm{SL}_{2m}$ ($m \geq 3$) which preserves a ...
- 7,300
14
votes
1
answer
1k
views
Expansion of $\det(A+B)$
If $A,B\in{\bf M}_n(k)$, then the following formula holds true:
$$\det(A+B)=\sum_{r=0}^n\sum_{|I|=|J|=r}\epsilon(I,I^c)\epsilon(J,J^c)A\binom IJ B\binom{I^c}{J^c}.$$
In this formula, $I$ and $J$ are ...
- 52.3k
2
votes
1
answer
88
views
Dubious matrix monotonicity
Coming from a problem in game theory, I arose at some dubious monotonicity like property for matrices of the following art. Let $H=\lbrace h\in\mathbb{R}^{n}\colon h_{1}+\dots+h_{n}=0\rbrace$. I'm ...
- 289
44
votes
2
answers
2k
views
Fermat's Last Theorem for integer matrices
Some years ago I was asked by a friend if Fermat's Last Theorem was true for matrices. It is pretty easy to convince oneself that it is not the case, and in fact the following statement occurs ...
- 1,889
8
votes
3
answers
663
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Representation theorem for matrices (reference request)
Motivation. If $A \in \mathbb{C}^{n \times n}$ is self-adjoint (or, more generally, normal), then we all know that
$$
A = \sum_{k=1}^n \lambda_k \, h_k \otimes h_k,
$$
where $\lambda_1,\dots,\lambda_n$...
- 12.5k
13
votes
2
answers
414
views
Is every finite-order unimodular matrix conjugate to a $0,1,-1$ matrix?
Problem. Given a matrix $A\in\mathrm{GL}(n,\mathbb{Z})$ such that $A^k=1$ for some $k\geq 1$, is there a matrix $g\in\mathrm{GL}(n,\mathbb{Z})$ such that $gAg^{-1}$ has only $0$, $1$, and $-1$ as ...
- 23.3k
1
vote
0
answers
49
views
Permutation of eigenvalues induced by a loop
A friend of mine just mention me what I think is a very fun phenomena and I would be very interested to learn more about it:
Let $A,B\in \mathbb{C}^{n\times n}$ two matrices. And let $\lambda_1(z), \...
- 4,361
3
votes
0
answers
231
views
Singularity of symmetric block matrix with singular diagonal blocks
One can show that the following statement holds:
Given symmetric matrix $A \in \Re^{n \times n}$ and tall matrix $B \in \Re^{n \times p}$ with full column rank,
$$\begin{bmatrix}A & B \\ B^T &...
- 73
7
votes
0
answers
133
views
Removing rows to reduce the rank
What is the smallest number of rows one can delete from a matrix to reduce its rank (by $1$)? Is there any standard name / notation for this characteristic? Has it been studied?
I am in fact ...
- 23k
2
votes
1
answer
1k
views
Is it faster to compute eigenvalues or coefficients of characteristic polynomials?
Given $A \in \mathsf{M}_n(\mathbb{C})$ (no special structure) is it (generally) faster to compute its eigenvalues or the coefficients of its characteristic polynomial?
References/insights would be ...
- 1,719
11
votes
1
answer
467
views
Correspondence between matrix multiplication and a graph operation of Lovász
In his book "Large networks and graph limits", Lovász describes a multiplication operation (he calls it concatenation) on "bi-labeled graphs". An $(m,n)$ bi-labeled graph is a ...
- 2,339
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
0
answers
359
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
1
vote
0
answers
187
views
Simultaneous diagonalization of the tensor products of Dirac gamma matrices
Let $\gamma_i\ (i=1,2,\ldots N)$ be the Dirac gamma matrices satisfying the Clifford algebra
$$\gamma_i\gamma_j+\gamma_j\gamma_i=2\delta_{ij} I\ \ (i,j=1,2,\ldots,N).$$
Then the tensor products $\...
- 241
5
votes
1
answer
514
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: $$M = \begin{bmatrix} a_{11} & 0 & \cdots & 0 & \cdots & & 0 & b_{1n} ...
- 169
1
vote
0
answers
158
views
Non-trivial ways for generating matrices $A$ for which $A + A^T$ is positive-definite?
Disclaimer: This might be an SE question, but I'm not quite sure...
Thanks in advance!
Setup
So, it is known (see Proposition 5.2) that if $A + A^T$ is positive-definite then $A$ must be a $P$-...
- 6,853
14
votes
1
answer
544
views
Are there nice isomorphisms $\operatorname{S}^2(k^n)\cong\Lambda^2(k^{n+1})$?
This might be forced to migrate to math.SE but let me still risk it.
The spaces $\operatorname{S}^2(k^n)$ and $\Lambda^2(k^{n+1})$ from the title have equal dimensions. Is there a natural isomorphism ...
- 18.4k
3
votes
0
answers
65
views
How to show that a continuous family of symmetric matrices is uniformly positive?
My problem : I have a family of $4 \times 4$ symmetric matrices. More precisely consider an interger $d$, a real $\lambda> 0$ and define the family $S_{\lambda}$:
$ \{A(\lambda,x_1,x_2) ; (x_1,...
- 31
5
votes
1
answer
1k
views
Spin groups in terms of matrices and/or linear operators
Thus far, the books and articles I have read dealing with spin groups $\mathbf{Spin}(n)$ and $\mathbf{Spin}(p,q)$ consider them only in terms of either Clifford algebras or topologically as the double ...
- 349
4
votes
2
answers
1k
views
Reference request: Oldest linear algebra books with exercises?
Inspired by the recent success of my "soft question" here, I also have to ask, what are some of the oldest linear algebra books out there with exercises? I'm fine with or without solutions, either way....
5
votes
2
answers
249
views
Eigenvalue density of a symmetric tridiagonal matrix
Let $A_n\in\mathbb{R}^{n\times n}$ be defined as
$$
A_n=\begin{bmatrix} a & b & 0 & \cdots & \cdots & 0 & 0\\ b & a & b & \cdots & \cdots & 0 & 0\\ 0 &...
- 2,712
3
votes
0
answers
62
views
How likely is a matrix with exactly $n$ number $1$s per row to avoid a large wide empty submatrix?
Consider the finite collection $M(N,n)$ of all $N \times N$ matrices with exactly $n$ entries per row equal to $1$ and all other entries equal to zero $0$.
By an $a \times b$ submatrix of $M$ we ...
- 1,955
4
votes
1
answer
372
views
Eigenvalues of random matrix conditional on positive definiteness
Consider the Gaussian Orthogonal Ensemble, considered as a probability measure $\mu$ on the space of real symmetric matrices. Let $\mu|PD$ denote this measure conditioned on the event that the matrix ...
- 505
4
votes
1
answer
172
views
Smith normal form and affine buildings
In Smith Normal Form of powers of a matrix someone has commented saying that one can reformulate many questions about Smith normal forms in the language of affine buildings. I wanted to know of a ...
- 41
2
votes
1
answer
341
views
Symmetric orthogonal matrices with constant diagonal entries
$\newcommand{\al}{\alpha}
\newcommand{\be}{\beta}
\newcommand{\de}{\delta}
\newcommand{\De}{\Delta}
\newcommand{\ep}{\varepsilon}
\newcommand{\ga}{\gamma}
\newcommand{\Ga}{\Gamma}
\newcommand{\la}{\...
- 128k
8
votes
1
answer
441
views
A question on symmetric matrices
$\newcommand{\R}{\mathbb{R}}$
The question is
Is there a constructive (say, parametric) description of the set (say $M_n$) of all symmetric matrices $A\in\R^{n\times n}$ such that all the diagonal ...
- 128k
4
votes
0
answers
98
views
Rank of binary matrix related to the number of positive squarefree integers less than $n$
I posted this question at the Mathematics SE, but received no response there so I am posting it here.
The following fact is stated in the comments-section of sequence A013928 in the OEIS.
Let $C$ ...
- 1,719
1
vote
0
answers
67
views
Classification of Maximal Rank Skew-Symmetric Matrices with Linear forms as entries
I am interested in canonical forms/simplifying assumptions of $(2n+1)\times (2n+1)$ skew symmetric matrices with homogeneous degree 1 polynomials in $k[x,y,z]$ as entries, and whose rank is $2n$.
I ...
- 553
1
vote
0
answers
619
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
5
votes
1
answer
473
views
higher order analogues of sylvester's law of inertia?
Sylvester's law of inertia (here I quote wikipedia)
If A is the symmetric matrix that defines the quadratic form, and S is any invertible matrix such that D = SAS^{T} is diagonal, then the number ...
- 305
1
vote
1
answer
925
views
What is special about 2 + $\sqrt{3}$?
Well, one thing is special about it, but it takes a while to explain.
Please let me know, whether this number occurs in other special occasions as well.
The explanation: Let $p$ be a complex ...
- 429
4
votes
1
answer
414
views
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 ...
- 143
3
votes
0
answers
82
views
Maximum number of negative entries in a matrix with positive diagonal and given rank
Suppose $A \in \mathbb{R}^{n \times n}$ has positive entries on it main diagonal and $\mbox{rank}(A) =: d < n$. Then, what is the maximum number of many negative entries $A$ can contain?
If, in ...
- 31
4
votes
0
answers
148
views
A Toeplitz variant of the Hilbert matrix
It is well-known that the Hilbert matrix $H$, i.e., the symmetric Hankel matrix with entries
$$H_{m,n}=\frac{1}{m+n-1}, \quad m,n\in\mathbb{N},$$
determines a bounded operator on $\ell^{2}(\mathbb{N}...
- 2,188
2
votes
1
answer
217
views
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}^...
- 3,625
3
votes
1
answer
449
views
Explicit formula for the functional calculus of 2x2 matrices
Wikipedia gives the following explicit formula for the functional calculus of $2\times2$ matrices:
$$
f(A) = \frac{f(\lambda_+) + f(\lambda_-)}{2} I + \frac{\mathrm{tr}(A)/2 - \mathrm{adj}(A)}{\sqrt{\...
- 7,127
11
votes
1
answer
632
views
How do computer algebra packages like Sagemath implement rank of a matrix
I am not sure if this is the right place to ask this question, but I believe there will be people here who do computations on computer algebra packages like Sage in their work.
I have been using ...
- 263
5
votes
1
answer
315
views
Rank-constrained least-squares solution of the Sylvester matrix equation
For the Sylvester matrix equation $AX+XB=C$, I want to find the least-squares solution $X$ via
$$\begin{array}{ll} \text{minimize} & \| AX + XB - C \|_{\text{F}}^2\\ \text{subject to} & \mbox{...
- 191
14
votes
4
answers
3k
views
Vandermonde matrix is totally positive
A totally positive matrix $M\in \mathcal{M}_{n\times m}(\mathbb R)$ is such that all of its minors of all sizes are positive. It is true that any Vandermonde matrix (with well-ordered positive entries)...
- 5,428
4
votes
0
answers
244
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
11
votes
1
answer
946
views
Detailed modern references for basic properties of Pfaffians over commutative rings
Pfaffians are important to algebraic combinatorics, at least.
This is to propose the making of a 'wiki' list, more modern, precise and compressed than e.g. the relevant Wikipedia page (nothing against ...
- 6,051
2
votes
0
answers
102
views
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 ...
- 121
7
votes
0
answers
197
views
A special eigenvalue problem
For my research I need to solve a generalised eigenvalue problem
$Ax=\lambda B x$, where $A$, $B$ are general matrices, and selectively find only eigen-pairs $\lambda, x$ such that $\lambda\in \mathbb{...
- 492
7
votes
1
answer
223
views
Result attribution for eigenvalues of a matrix of Pascal-type
A few years ago, I wanted to cite a result in a paper, for which I could not find a reference. I ended up not using the full strength of it, and the part that I needed could be easily proved. Still, I'...
- 1,190
3
votes
1
answer
319
views
Matrix transformation that "rotates" a matrix by $45^\circ$
I have an $n \times n$ integer matrix $A$. I want to obtain an $m \times m$ matrix $B$, where $m \ge n$, such that the rows of $A$ are the diagonals of $B$ and the columns of $A$ are the anti-...
- 2,993