All Questions
93 questions
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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 ...
- 51
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
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}^...
- 22.5k
8
votes
2
answers
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 $T(A,B)=AB$. ...
- 31
11
votes
1
answer
633
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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
$$\begin{array}{ll} \text{minimize} & \| AX + XB - C \|_{\text{F}}^2\\ \text{subject to} & \mbox{...
4
votes
0
answers
245
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On the sum of the first row of the inverse of a certain symmetric Toeplitz matrix
(i) Consider a Toeplitz matrix $A_n = (a_{i, j})_{1 \le i, j \le n}$ of size $n$ defined as follows:
$$ a_{i, j} := |i-j|^{-1/2}, \text{ if } i \ne j; \ \ a_{i, j} := 2, \text{ if }i = j. $$
Let $...
- 195
2
votes
0
answers
102
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Eigenvalue distribution for a real-valued random matrix with correlated Gaussian entries
I'm working on an application where I would greatly benefit from knowing the distributions of the eigenvalues of a real-valued random matrix whose elements can be assumed to be Gaussian, but where I ...
- 321
10
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3
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830
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Find the inverse of a matrix that is very similar to the Hilbert matrix
The standard Hilbert matrix $H$ is given by
$$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
views
Determinant and inverse of a "stars and stripes" matrix
This is a variant of another MO question. Consider the matrix
$$M_n:=\begin{bmatrix}c_1& a & b&a& \ddots & a \\ b & c_2 & a& b&\ddots & b\\ a & b & c_3&...
CommunityBot
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5
votes
2
answers
389
views
Pfaffian of several skew-linear transformations / matrices
Introduction: Let's assume we have a 2-form $\alpha=(1/2)\sum_{j,k=1}^n a_{jk}\ e_j\wedge e_k$, where $n=2m$, and $a_{jk}\in\mathbb C$. We know that $\alpha^{\wedge m}=\alpha\wedge\alpha\dots\wedge\...
- 8,597
1
vote
0
answers
47
views
Partial ordering of a matrix entries [closed]
I need this for experimentation in some work, so it is not without purpose.
Consider the in-spiraling and out-spiraling $4\times 4$ matrices
$$\begin{pmatrix} 1&2&3&4 \\ 12&13&14&...
- 43.2k
1
vote
1
answer
111
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Decomposition of integral non-generate matrices [closed]
Let $GL_{\eta}(n,\mathbb{Z})=\left\{a\in
GL(n,\mathbb{R})\cap M^{n\times n}(\mathbb{Z})|det(a)=\eta\right\}$. Prove that there exists a
finite number of matrices $a_i$ in $GL_{\eta}(n,\mathbb{Z})$ ...
- 333
8
votes
2
answers
2k
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Bounding the minimum singular value of a block triangular matrix
Question:
What is the sharpest known lower bound for the minimum singular value of the block triangular matrix
$$M:=\begin{bmatrix}
A & B \\ 0 & D
\end{bmatrix}$$
in terms of the properties ...
CommunityBot
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11
votes
3
answers
918
views
yet another determinant and inverse of a matrix
This problem is some variation of another MO question. Consider the matrix
$$M_n:=\begin{bmatrix}-c& a & a& \dots & a \\ b & c & a& \ddots & a\\ b & b & -c &...
CommunityBot
- 1
1
vote
2
answers
285
views
a follow up question on traces of matrices
In a recent MO post, pallab1234 ask for trace inequalities for which counterexample were given. I wish to probe in a different direction.
Suppose $A, B$ are $n\times n$ symmetric matrices (with ...
CommunityBot
- 1
5
votes
1
answer
199
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Find the inverse of a more general matrix that is similar to the Hilbert matrix
In the last MO question , the following matrix is given:
$$M_{ij}=\left[\frac{1+(-1)^{i+j}}{i+j-1}\right]$$
and its inverse has been discussed.
Now the problem is further extended to a more general ...
CommunityBot
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15
votes
4
answers
4k
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Kernel of skew-symmetric matrix of rank $n-1$ with $n$ odd: is this a known result?
When $n$ is odd, the kernel of a skew-symmetric matrix $M$ of size $n\times n$ and rank $n-1$ is the span of $v$, where $v$ is a vector whose $i$-th component is the Pfaffian of the matrix obtained by ...
- 52.3k
3
votes
1
answer
655
views
Upper bounds on the condition number of the eigenvector matrix
Let $A$ be an $n\times n$ real matrix with entries in a fixed interval $[a_\min,a_\max]$, with $a_\min$, $a_\max>0$.
Question: Are there any upper bounds on the condition number of the ...
- 2,286
10
votes
1
answer
520
views
Homogeneous polynomials, mixed determinants, positive definiteness
Are there $n\times n$ real matrices $A_{1}, \ldots, A_{n}$ such that the $n$-homogeneous polynomial
$$
f(x_{1}, \ldots, x_{n}) = \det(x_{1} A_{1}+\cdots +x_{n} A_{n})
$$
never vanishes on $\...
- 109k
13
votes
1
answer
1k
views
An inequality for the spectral radius of matrices used by J. Bochi
I am interested in the history of an inequality for the spectral radius of a $d\times d$ real or complex matrix, which occurs in Jairo Bochi's 2002 article Inequalities for numerical invariants of ...
- 6,206
10
votes
0
answers
477
views
Name for an operation on matrices?
Given two matrices $A$ and $B$ of size $a \times n$ and $b \times m$ consider the following operation $A \dagger B$ whose result is an $a b^n \times n m$ matrix. $A \dagger B$ is a block matrix with $...
- 33.8k
0
votes
1
answer
229
views
Reference request: Strong Connectivity and the Incidence Matrix
Question: What would be a good reference for characterizations of strong connectivity of a digraph in terms of its incidence matrix?
Details: Consider a digraph $(V, E)$ with vertex set
$$V = \{v_1,...
CommunityBot
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1
vote
1
answer
250
views
characterize certain type of matrices
I am trying to characterize matrices with a certain property :
Define $U$ as an $n \times n$ matrix (over C or R; you can also assume
that it is unitary or orthogonal if it helps). Now take $n$
...
- 443
6
votes
1
answer
192
views
Monte-Carlo computation of the Smith normal form
Quite some time ago I saw an article where a Monte-Carlo algorithm for computing the Smith normal form of an integer matrix was described. In this article the following problem was posed:
Suppose $P, ...
- 39.9k
7
votes
2
answers
251
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What methods do we have to understand the spectrum of matrices with restricted entries?
Consider questions of the form (or the "most probable value of" version of these questions rather than the "largest possible"),
What is the largest possible spectral radius of a $...
CommunityBot
- 1
0
votes
1
answer
130
views
Reference for measures of commutativity needed
I'm looking for an appropriate measure to quantify the extent to which two matrices commute. In other words, if A and B are two n×n Hermitian matrices, and [A,B]=C.
I'd like a function μ:Cn×n→[0,∞) ...
- 3,741
5
votes
0
answers
254
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A weak Perron-Frobenius property for sets of positive matrices
A popular form of the Perron-Frobenius theorem states the following result: if $A$ is a $d \times d$ real matrix all of whose entries are positive, then the spectral radius of $A$ is a simple ...
- 6,206
2
votes
1
answer
336
views
Bringing a (Least Squares Problem) Matrix into Block Upper-triangular Shape via Matrix-reordering
I have the problem of solving very large and very sparse least squares problems and, a bit dissatisfied with the run-times of the full-fledged QR-algorithm, I would like to bring the instances into ...
CommunityBot
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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 ...
- 12.7k
3
votes
3
answers
1k
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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 \...
- 1,897
2
votes
3
answers
1k
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On certain decomposition of unitary symmetric matrices
This is by any means elementary, but since I have asked this question on Stark Exchange but received no satisfactory answers I decide to post it here.
It is well known that a symmetric matrix over ...
- 1,759
5
votes
1
answer
2k
views
Rank of a 0-1-matrix
Suppose $K$ is a field of characteristic $0$. Let $M \in K^{n \times m}$ be a matrix such that every entry of $M$ is either $0$ or $1$. About this matrix, I know further that each sum over a column ...
- 6,210
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 ...
- 4,994
29
votes
6
answers
10k
views
how to find/define eigenvectors as a continuous function of matrix?
I asked this (with background) here
https://stats.stackexchange.com/questions/38494/principal-component-analysis-bootstrap-and-probability-of-eigenvalue-collision
but did not really get any answers. ...
CommunityBot
- 1
1
vote
1
answer
417
views
Decomposition of Matrix to its sub-matrix with constant rank
When we study the structure of simple graphs with a lot of $1$ or $-1$ as its adjacency eigenvalues, the rank of its adjacency matrix is very important. The reason is, in these case, we can study the ...
- 54.2k
16
votes
3
answers
791
views
Random products of projections: bounds on convergence rate?
The von Neumann-Halperin [vN,H] theorem shows that iterating a fixed product of projection operators converges to the projector onto the intersection subspace of the individual projectors. A good ...
- 201
1
vote
1
answer
254
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references for families of conditionaly negative definite matrices
We say that a matrix $A\in M_n(\mathbb{C})$ is a conditionaly negative definite matrix if it is hermitian and if for all complex numbers $c_1,\ldots,c_n$ such that $c_1+\cdots +c_n=0$ we have
$$
\sum_{...
CommunityBot
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3
votes
1
answer
621
views
Largest eigenvalue of a periodic Jacobi matrix
There is a vast literature on Jacobi matrices, I just don't know where to start looking. I'm interested in estimating the largest eigenvalue of the $n\times n$ periodic Jacobi matrix $D+P+P^{-1}$, ...
- 3,343
6
votes
0
answers
998
views
Generalized Courant-Fischer theorem
Consider some quaternionic matrix $A$. A right eigvenvalue of $A$ is a quaternion $q$ such that $Ax=xq$ for some $x\in \mathbb{H}^n$. Similarly, a left eigenvalue of $A$ is quaternion $q$ such that $...
- 475
16
votes
4
answers
3k
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How many minors I need to check to conclude all minors will vanish ?
Given a $m \times n$ matrix $n>m$, I was trying to check if all its $m \times m$ minor vanish.
I remember hearing that one really does not need to check all possible minors in order to conclude ...
- 10.7k
4
votes
1
answer
1k
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Integer vectors in the kernel of an integer matrix
Let $A$ be a non-zero symmetric $n \times n$-matrix with integer entries and suppose that $\det(A) =0$.
Question: How long is the shortest non-zero integer vector in the kernel of $A$?
Example: If ...
CommunityBot
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2
votes
2
answers
3k
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Statement of Lagrange's theorem on determinants(elementary question).
Apologies for this elementary question; but I was unable to find a reference otherwise.
Let $A, B, C$ be square matrices of the same dimension. Then,
$$\begin{vmatrix} A & C \\\ 0 & B \end{...
- 33.8k