All Questions
6,291 questions
4
votes
1
answer
886
views
best rank r approximation for non-Frobenius norm
The best rank $r$ approximation to a given matrix $M$ in Frobenius norm, according to Eckart-Young theorem, is truncated SVD - just keep $r$ largest singular values. What if I need to construct best ...
- 263
23
votes
0
answers
8k
views
An $n \times n$ matrix $A$ is similar to its transpose $A^{\top}$: elementary proof?
A famous result in linear algebra is the following.
An $n \times n$ matrix $A$ over a field $\mathbb{F}$ is similar to its transpose $A^T$.
I know one proof using the Smith Normal Form (SNF). ...
- 3,320
3
votes
0
answers
75
views
Are there a group of mappings from (n-1)-dim space to an (n-1)-sphere guaranteeing the orthogonality of images?
Hello, everyone.
As we know that in an $n$-dimensional Euclidean space $\mathbb{R}^n$, there exists a continuous bijective mapping from a subset $V^{n-1}\subseteq\mathbb{R}^{n-1}$ to a unit $(n-1)$-...
- 607
3
votes
1
answer
5k
views
Sum of elements of inverse matrix
Hello all,
Assume NxN matrix A of complex values. I want to calculate the sum of all elements of its inverse.
The problem is that calculating the inverse is computationally expensive and since I am ...
0
votes
2
answers
1k
views
Is it possible to obtain the vectors orthogonal to a given one by orthogonal transformations?
Hello, everyone!
Supposing that there is a unit vector in $n$-dimensional real space $\mathbf{x}_1\in\mathbb{R}^n$, I want to get a group of $n-1$ vectors to form an orthogonal basis with $\mathbf{x}...
- 607
17
votes
1
answer
2k
views
Subgroups of $\mathbb{Z}^n$
I hope that the following problem isn't actually elementary (at least, for the sake of the fact that I'm posting it here), and I apologize if it is. I did try hard to solve it first.
Let $V$ be a $\...
- 1,786
11
votes
4
answers
3k
views
Classification of Tori of GL2, up to conjugation
Over an algebraically closed field $k$, every one-dimensional torus embedded (as a closed algebraic subgroup) into GL2 is diagonalisable, and the embedding is $t\mapsto (t^m,t^n)$ for some integers $m,...
- 7,722
7
votes
0
answers
2k
views
Computing the characteristic polynomial without determinants
Given this matrix:
$\\\
A=\left( \begin{array}{cccccccc}
3 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\\
1 & 0 & 0 & 0 & -1 & 0 & 0 & 0 \\\
0 & 3 & ...
- 113
2
votes
0
answers
86
views
Most efficient algorithm for computing norm of the residual for the least squares problem in the rank deficient case
I have a large $m\times n$ data matrix $A$, $m>n$, and response $m$-vector $b$. I need to calculate $E = ||Ax-b||_2$ as quickly as possible, where $x$ is the least squares solution. I don't need ...
- 21
4
votes
1
answer
1k
views
How do minimal polynomials relate?
How does the idea of a "minimal polynomial" for a matrix (i.e. for a matrix $A$, the polynomial, $\mu (x)$, of least degree, such that $\mu (A) =0$) relate the the "minimal polynomial" for some ...
- 113
13
votes
1
answer
2k
views
Number of idempotent $n\times n$ matrices over $\mathbb{Z}/m\mathbb{Z}$?
Is there any known formula for the number of idempotent $n\times n$ matrices over $\mathbb{Z}_m:=\mathbb{Z}/m\mathbb{Z}$ ?
The number of idempotent matrices over a finite field is well-known and ...
user31416
6
votes
3
answers
1k
views
Subspace of Skew-symmetric Matrices of Rank Four
Let $n\geqslant 5$ and let $E_4(n)$ be a linear subspace of $(n\times n)$- real skew-symmetric matrices such that
$$
rank(A)=4,\text{ for all }A\in E_4(n),A\neq 0.
$$
I'm curious about the following ...
- 895
1
vote
0
answers
191
views
Eigenvectors of contraction times projection
Suppose $A$ is a real $n\times n$ matrix with real eigenvalues:
$$
1=\lambda_1>|\lambda_2|\ge \ldots\ge |\lambda_n|>0.
$$
Suppose $B$ is an involution, for simplicity let us assume that
$B$ is ...
- 650
0
votes
0
answers
270
views
Solution Existence of a System of Complex Quadratic Equations
Consider $ {x_k}_{k = 1}^K \in \mathbb{C}^{N \times 1}$ a set of $K$ complex vector variables of length $N$. I am interested in finding the existence of a solution of the following
quadratic set of ...
- 342
1
vote
0
answers
107
views
Complementation in an extension field
If $E$ is an extension field of $F$, is $F$ necessarily (without assuming the axiom of choice) complemented as a vector subspace of $E$? (Of course the answer is easily yes if the extension is finite....
- 11.4k
3
votes
0
answers
172
views
Semirings where solving linear systems is in P
Solving linear systems appears hard in semirings.
In $\mathbb{N}_0 (+,\times)$ it is NP-complete via reduction to 1-in-3 SAT.
In the min-plus semiring the complexity is $ NP \cap coNP$ according to ...
- 25.4k
3
votes
3
answers
793
views
Find a convex hull that contains given points?
Suppose that you have vectors $a_{1},...,a_{m}$ in $\mathbb{R}^n$. Can you take $n$ among them, let $v_{1},...,v_{n}$ such that $a_{1},...,a_{m}$ belong to the convex hull of $(cn)v_{1},-(cn)v_{1},...,...
user31317
2
votes
2
answers
421
views
On matrix norms
It is standard to define an induced matrix norm $|||\cdot|||$ from a vector norm $||\cdot||$ in this way:
$|||A|||=\max_{x \neq 0}{\frac{||Ax||}{||x||}}$.
Suppose we define a different function of ...
- 6,962
4
votes
1
answer
1k
views
Matrix perturbation theory
I am having matrix $M_0$ with coresponding eigenvectors and 4 eigenvalues {0,0,a,-a}. Eigenvalue $\lambda=0$ is double degenerated. Now I am appliing small perturbation $\epsilon M_1$ and want to get ...
- 41
0
votes
1
answer
139
views
Spectrum of a Laplacianized matrix
Suppose that $A$ is a positive matrix and that we let $R$ be the diagonal matrix of $A$'s row-sums. What can be said about the spectrum of $R-A$? I am particularly interested in the largest eigenvalue ...
- 6,962
4
votes
1
answer
543
views
Rank of a matrix with missing entries
Let $M$ be a $2^n \times 2^n$ matrix over real number field, where the rows and columns are indexed by subsets of $[n] := \{1,2,\ldots,n\}$, and defined as follows,
$
M_{A, B} = 1
$
if $A \subseteq B$;...
- 651
2
votes
0
answers
136
views
Possible restrictions on generators of $M_n(\mathbb{C})$
Suppose matrices $a$ and $b$ generate $M_n(\mathbb{C})$. I would like to know what restrictions this imposes on $a$ and $b$. More concretely, do there exist $a,b\in M_n(\mathbb{C})$, which generate $...
- 179
0
votes
1
answer
1k
views
number of non-negative integer solutions for a set of equations [closed]
How to find the exact number of non-negative integer solutions of the following set of equations :
$$x_1 + x_2 + x_3 + x_4 + x_5 + x_6=6 $$
$$ 2x_1 + x_2 + x_3 = 4$$
$$ x_2 + 2x_4 + x_5 = 4$$
$$ x_3 +...
- 209
2
votes
2
answers
175
views
linear independence of orbits via a set of transformations in char p
Let $T_1, \ldots, T_n \in GL(n,\mathbb{F}_p)$. Suppose for all $\vec{v} \in \mathbb{F}_p^n$ we have $\det (T_1 \vec{v}, T_2 \vec{v}, \ldots, T_n \vec{v}) = 0$. Now, let $k$ be a finite extension of $\...
- 23
1
vote
1
answer
324
views
Linear algebra and Cayley Hamilton
Consider the transform $\widehat{b}=T^{-1}b$, where $T=\begin{bmatrix}b & Ab & A^2b & \dots & A^{n-1}b \end{bmatrix}$ has full rank. Is it possible to find an explicit expression for
\...
- 13
17
votes
1
answer
4k
views
Geometric interpretations of matrix inverses
$A$ is an invertible $n \times n$ matrix. Interpret each row of $A$ as a point in $\mathbb{R}^n$; then these $n$ points define a unique hyperplane in $\mathbb{R}^n$ that passes through each point (...
- 693
4
votes
1
answer
197
views
Lower bound on the dimension of a subspace of $\mathbb Z_2^r$?
This question may be very trivial, I apologize if it is so.
I have subspace $V\subset \mathbb Z_2^r$ with the property that for every choice of a subset $I$ of $k$ elements in $\{1,2,\dots r\}$, the ...
- 6,253
8
votes
2
answers
1k
views
A basis for Schur functors
Suppose $V$ is a finite-dimensional vector space (over $\mathbb{C}$) and $\lambda$ is a partition of $n$ (not necessarily the dimension). Let $S^\lambda(V)=(V^{\otimes n})_\lambda$ be the $\lambda$'th ...
- 7,962
1
vote
0
answers
192
views
Non-negative Quadratic forms with Exterior Forms
Hello All,
I apologize if the following question is too elementary. Any suggestion is greatly appreciated. Thank you.
Let $n\geqslant 4$, $X$ be an $n$-dimensional inner product space over $\mathbb{...
2
votes
1
answer
247
views
Factorization of bivariate polynomial
Let $q(y, z) = u_1 + u_2y + u_3 z + u_4y^2 + u_5yz + u_6z^2 + u_7y^3 + u_8y^2z + u_9yz^2 +$ $\hspace{2.55cm}u_{10}y^3z + u_{11}y^2z^2 + u_{12}y^3z^2$
Can $q(y, z)$ be factorized as
\begin{...
- 23
0
votes
2
answers
573
views
about decomposition of three forms
Patrick D. Baier in his Ph.D. thesis for proving the theorem 2.1.4 used the following non-trivial fact (in chapter 2 on page 14):
Let $0\neq X\in V$ (here $V$ is of dimension 6), $W^\ast = Ann(X)$ ...
user21574
7
votes
2
answers
943
views
Hilbert's Nullstellensatz on polynomials with integer coefficients
Let $f_1, f_2, \ldots, f_m \in \mathbb{Z}[x_1, \ldots, x_n]$. Assume $f_1(X) = f_2(X) = \ldots = f_m(X) = 0$ have no solutions over $\mathbb{C}^n$, then by Hilbert's Nullstellensatz, there exists ...
- 651
1
vote
2
answers
865
views
How to identify bridge nodes between nearly connected graph components in partitioned adjacency matrices?
I have adjacency matrices which have nearly connected components. That is partitions with a dense number of edges between nodes in the same group and few edges acting as bridges between these groups. ...
- 197
2
votes
1
answer
1k
views
Subgradient of Minimum Eigenvalue
Consider three $N \times N$ Hermitian matrices $A_0$, $A_1$, $A_2$. Consider the function
\begin{align}
f(t_1,t_2)=\lambda_{\text{min}}(A_0+t_1A_1+t_2A_2)
\end{align}
where $\lambda_{\text{min}}$ ...
- 1,421
4
votes
1
answer
635
views
Full-rank linearly independent matrices
Can we find $n^2$ full-rank matrices in $\mathbb{F}^{n \times n}$ which are linearly independent (i.e. when vectorized are linearly independent)? If not, how many such matrices can be found?
- 91
0
votes
1
answer
2k
views
Find edge weights that fit given node weights
Let $G = (V,E)$ be a connected simple graph (unweighted, undirected, no selfloops) on $n$ nodes.
Let $\mathbf{d} := (d_1, d_2, ..., d_n) \in \mathbb{R}_{>0}^n$ be a vector of arbitrary given node ...
- 340
9
votes
0
answers
464
views
An identity for Hankel determinants
Is the following result about Hankel determinants known or a simple consequence of some known results?
Let
$f(x) = \frac{\displaystyle 1}{{\displaystyle 1 - \frac{{a x^{m + 2}}}{\displaystyle {1 - \...
- 5,622
1
vote
2
answers
502
views
When is PSU(2,q^2) = PSL(2,q) ?
The context for this question is from page 284 - 287 of Berger's paper: http://pdn.sciencedirect.com/science?_ob=MiamiImageURL&_cid=272332&_user=209810&_pii=S0021869398976785&_check=y&...
- 10.7k
4
votes
2
answers
958
views
How to write down explictly the isomorphism of two finite dimensional representation of compact groups?
When dealing with finite dimensional representations over $\mathbb C$ of a compact group $G$, Character Theory provides us with a convenient way to determine whether two representations are isomorphic....
- 41
1
vote
0
answers
290
views
Injective Mapping
Let y=Ax. A is a matrix n by m and m>n. Also, x gets its values from a finite alphabet. Elements of A and x can be complex numbers. How can i show if the mapping from x to y is injective for given A ...
- 11
15
votes
1
answer
1k
views
Free subgroups of $\mathrm{GL}(2,\mathbb{Z})$
Is there a bound $B$ such that every 2-generator subgroup
$G = \langle a, b \rangle \le {\rm GL}(2,\mathbb{Z})$
whose generators do not satisfy a relation of length $\leq B$ is free?
If it exists, ...
- 19.6k
3
votes
1
answer
413
views
Kronecker-structured matrix kernel
Let $A,B\in\mathbb{C}^{n\times 3n}$ be two matrices, and denote the Kronecker matrix product by $\otimes$. The matrix
$$
M=
\begin{bmatrix} A \otimes I_n \\\\ I_n \otimes B\end{bmatrix}
$$
has size $...
- 20.2k
0
votes
0
answers
257
views
What is the integer form of a projector into the intersection of the ranges of two integer projection matrices?
Consider two square integer matrices $X$ and $Y$ of the same dimension with the following properties:
$X^2=rX$, and $Y^2=sY$ for integers $r$ and $s$. The $\gcd$ of the entries of $X$ is 1 and the $\...
4
votes
0
answers
170
views
Decomposition of projectors: A generalized format
Let $V=\mathbb C^n$ be a vector space with (linear) maps $P_1:V\rightarrow V$ and $P_2:V\rightarrow V$ that are projectors , i.e. they satisfy $P_i^2=P_i$.
It is not hard to understand the structure ...
- 195
0
votes
0
answers
155
views
Convexity of a Certain Set of Covariance Matrices
Hello,
My question is about a certain set of matrices being convex or not. I'll start with some preliminaries in order to define myself properly. Let $X_1,U,X_2$ be three zero-mean Gaussian random ...
- 155
1
vote
0
answers
296
views
Finding lower triangular matrix of an indefinite matrix
So I have the system $M = RS = RQQ^{-1}S $ and I have $R$ and $S$ currently.
I impose some constraints on $R$ in the form of $r^T$$QQ^Tr = 1$ where $r$ and $r^T$ are rows of R and their transposes. ...
- 11
0
votes
0
answers
151
views
Ratio of Eigen values and Mutual Independence
Given a matrix $X$. Calculating the Eigen values of $XX^T$ and using the ratio of maximum and minimum eigen values normally gives the condition number of the matrix.
If $X$ contains $M$ observations ...
- 1
6
votes
2
answers
1k
views
Systems of simultaneous real quadratic equations
Starting from a problem in spectral graph theory, I got dragged into a problem in combinatorial matrix theory about constructing $n\times n$ real orthogonal matrices with a specified pattern of zero/...
3
votes
0
answers
119
views
Asymptotics of system of linear equations
This is a cross-post with minor edits of the unanswered question https://math.stackexchange.com/questions/269904/pair-of-recurrence-relations.
I have a system of equations as follows.
$$M(p) = 1+\...
- 31
13
votes
1
answer
516
views
Permanent of a matrix of odd integers
It is clear that the permanent of an $n\times n$ matrix which entries are odd integers, is an even number, as it is the sum of $n!$ odd numbers. I am interested in finding the highest power of $2$ ...
- 273