Questions tagged [linear-algebra]
Questions about the properties of vector spaces and linear transformations, including linear systems in general.
5,681
questions
5
votes
0
answers
108
views
Inverses of the sums of all possible subsets of a set of symmetric and positive definite matrices
I have a set of $c$ matrices $A_1 ... A_c$ which are all symmetric and positive definite. I would like to calculate the inverses of all the possible sums, i.e.
$(A_1+A_2)^{-1},(A_1+A_3)^{-1},(A_2+A_3)...
- 71
2
votes
1
answer
484
views
Classification of pairs of commuting endomorphisms
Let $K$ be an algebraically closed field. I'm interested in isomorphism classes of triples $(V,f,g)$ where $V$ is a finite dimensional $K$-vector space and $f,g$ are commuting endomorphisms of $V$. ...
- 7,199
0
votes
1
answer
234
views
does the basis in the singular value decomposition of a sum depend on the singular values of the summands
Suppose you have 4 matrices with singular value decompositions
$A = U_1 \Sigma_A V_1^{\dagger}$, $B = U_2 \Sigma_B V_2^{\dagger}$, $C = U_1 \Sigma_C V_1^{\dagger}$ and $D = U_2 \Sigma_D V_2^{\dagger}$ ...
20
votes
3
answers
795
views
Basis removal gives a basis
Let $V$ be a vector space. Let us say that a finite set $X$ of vectors in $V$ is harmonic if for $B \subseteq X$,
$$
B \text{ is a basis of } V \implies X \setminus B \text{ is a basis of }V.
$$
Let ...
- 3,912
0
votes
1
answer
100
views
The influence of eigendecomposition on the periodicity of a (rank 2) Hermitian matrix (of functions)
Let $\boldsymbol{R}(u,v);~u,v\in\mathbb{R}$ be a Hermitian matrix (of Hermitian functions) with entries
\begin{equation}
r_{ij}(u,v) = 1 + Ae^{-2\pi i \phi_{ij}(ul_0 + vm_0)}; A\in\mathbb{R},l_0\in\...
- 33
1
vote
1
answer
159
views
Finding the most compact representation of a vector in an "overdetermined base"
I want to find the most compact representation of a vector as a linear combination of a set of vectors B. B has more elements (on purpose) that is needs to have to describe the subspace.
For example
...
- 113
3
votes
1
answer
753
views
Expected number of random binary vectors so that the form a basis
I would like to compute the expected number of vectors in $\mathbb{F}_2^n$ we need to draw (following a uniform distribution) so that they form a basis of $\mathbb{F}_2^n$, i.e., that we have $n$ ...
- 43
3
votes
0
answers
693
views
Row subset selection of matrix to optimize condition number
Given a matrix $\mathbf{A} \in \mathbb{C}^{N\times M}$ with $N \gg M$. This matrix results from a linear equation system and has a certain structure (however, I do not think that details provide any ...
- 167
3
votes
0
answers
806
views
A problem on graph theory and complex numbers!
Let ${\mathcal G} = ({\mathcal V},{\mathcal E})$ be a simple connected undirected graph with $n$ vertices. Also let $z_1, \ldots, z_n \in {\mathbb C}$ be complex numbers such that
$$
||z_1||=\ldots = |...
8
votes
2
answers
405
views
Dimension of commutative subalgebras of a central simple algebra
let $k$ be a field, and let $A$ be a central simple $k$-algebra over $k$.
What is the maximal dimension of a commutative $k$-subalgebra of $A$?
If $A=M_r(D)$, where $D$ is a central division $k$-...
- 81
6
votes
2
answers
947
views
Minkowski successive minima inequality for a lattice base?
Let $\Lambda$ be a lattice of $\mathbb{R}^n$, and $\lambda_i$ be the radius of the smallest ball containing $i$ linearly independent lattice vectors.
The Minkowski successive minima inequality says ...
- 403
1
vote
0
answers
762
views
Determine lattice basis from given lattice points
I'm working on the Shortest Lattice Vector Problem (SVP) for a paper that I'm currently writing. I wish to verify whether a particular structural, namely the building block property ( refer to the ...
- 11
4
votes
1
answer
429
views
Finding all local maximum points of a function?
Let ${\boldsymbol \theta}=(\theta_1,\theta_2,\ldots,\theta_n) \in{\mathbb T}^n$ and $P:{\mathbb T}^n\rightarrow {\mathbb R}$ be a function defined on $n$-torus as
$$
P({\boldsymbol \theta}) = \sum_{i&...
1
vote
1
answer
176
views
Embedding of Two Objects Into Higher Dimensions With Their Sum
Given two vector sets, $\vec x_i$ and $\vec y_i$ (for $i$=1,2,...N, but the dimensionality of each vector can be more than N), let their sum set be $\vec z_i = \vec x_i + \vec y_i$. It's easy to ...
- 1,537
3
votes
1
answer
588
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 ...
- 7,218
6
votes
1
answer
4k
views
Taylor expansion of a function of a matrix
Assume that ${\mathbf H}$ is a $N \times M$ matrix. The following parameter is called orthogonality deficiency and describes how much orthogonal the columns of ${\mathbf H}$ are.
$$ od({\mathbf H}) ...
- 273
10
votes
2
answers
2k
views
How to transform matrix to this form by unitary transformation?
Without loss of gernerality, we can only consider $n$-dimensional diagonal matrix $M$ whose elements are all nonnegative, i.e.
$$M=\operatorname{diag}(m_1,m_2,\cdots,m_n)\ (m_i \geq 0).$$
Then is ...
- 273
3
votes
1
answer
412
views
Maximal size of an almost-disjoint linearly independent family in $K^{\mathbb{N}}$
Let $K$ be a field, say infinite, and denote by $L$ the $K$-vector space $K^{\mathbb{N}}$. What is the maximal cardinality of a $K$-linearly independent subset $X$ of $L$ such that any two distinct ...
- 76
2
votes
0
answers
46
views
Bases witnessing orthogonality of maps
Say $A$ and $B$ are finite-dimensional inner product spaces, and $L$ and $M$ are linear maps from $A$ to $B$.
If there are orthonormal bases $\mathcal{A}$ for $A$ and $\mathcal{B}$ for $B$ such that ...
- 2,224
2
votes
3
answers
344
views
Geometric means of matrices beyond the positive definite cone
Recently a lot of work has been done on geometric means of positive definite matrices (see here and here for example). Has anyone extended this concept to larger sets of matrices (copositive, for ...
- 6,882
3
votes
1
answer
1k
views
What is the significance of matrix ordered algebras?
I am trying to grok matrix ordered operator algebras, but I am having a hard time understanding their significance from the definition. Here is the definition (or at least, one way of stating it):
...
1
vote
2
answers
1k
views
Alternative to Choleski Decomposition for Correlation Matrix
Let $\Sigma$ be a correlation matrix, ie. symmetric. The Choleski decompositon gives upper triangular $A$ such that $A^TA = \Sigma$. Instead of upper triangularity, we are looking for $A$ that is not ...
- 11
2
votes
1
answer
899
views
Paralel bezier curve
If I have a cubic Bezier curve specified by two endpoints and two control points, how can I find an offset curve which is "parallel" to the original at some given distance, after i have determined the ...
- 123
3
votes
1
answer
341
views
Iterated Lefschetz numbers
Given a pseudo Anosov mapping class $f:S_{g,n}\rightarrow S_{g,n}$ is the Lefschetz number for $f^m$ negative for some $m$ depending only on $(g,n)$?
The Lefschetz number of a mapping class $f$ can ...
- 105
2
votes
1
answer
1k
views
Iwasawa Decomposition for Matrices [closed]
I was asked to prove that if
$$ T_{n}^{+}(\mathbb{R}) \subseteq M_{n}(\mathbb{R})$$
denotes the set of upper triangular matrices with positive diagonal entries, then prove that the multiplication ...
- 339
1
vote
0
answers
196
views
A generalization of strictly upper triangular matrces
Let $A = [a_{ij}]_{n\times n}$ be a real matrix with the property $a_{ij}a_{ji} = 0$. What can be said about the eigenvalues of$A$ ? I want to know when $A$ is non-singular and when $A$ is nilpotent. (...
- 191
6
votes
1
answer
763
views
a question of ranks of matrices
Let $S=\{A_{1}, ..., A_{m}\}$ be a set of $n \times n$ symmetric matrices and $m>n$, the rank $r(A_{i})=1$ for each $i$. Suppose that for any $m-1$ matrices $\{A_{i_{1}},...,A_{i_{m-1}}\}$ in $S$, ...
- 1,911
5
votes
1
answer
223
views
Uniform hyperbolicity decay estimate
I have been trying to understand the proof of the following result, which is considered well-known.
Theorem: Fix a compact metric space $X$, a homeomorphism $T:X \to X$, and a continuous map $ A : ...
- 437
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
5
votes
1
answer
458
views
A naive question on eigensheaves for group actions on derived categories
In this Mathoverflow question, Examples of Eigensheaves outside of langlands, David Ben-Zvi says
" Given a G -space X you can recover quasicoherent sheaves on X from sheaves on X/G (ie equivariant ...
- 1,331
1
vote
1
answer
501
views
Semidefinite relaxation for a quadratic feasibility problem using CVX
The following decides the feasibility of a semidefinite program (SDP)
\begin{align}
\max_{\mathbf{Z}}~0 \\\
\mathrm{trace}(\mathbf{Z})\leq \rho \\\
\mathrm{trace}(\mathbf{S}_1\mathbf{Z}) \geq \alpha \...
- 1,371
7
votes
3
answers
295
views
Sets from $(F_2)^n$ which are not fixed by any non-identity isomorphism
This is a followup question to the discussion in the comments of
Sets which are not fixed by any non-identity isomorphism
So consider a finite $n$-dimensional vector space $V$ over $F_2$. For which ...
- 407
8
votes
3
answers
460
views
Sets which are not fixed by any non-identity isomorphism
Consider a finite dimensional vector space $V$ over a field (finite or infinite but big enough). I am looking for a subset $W$ of $V$ such that for any bijective but non-identity linear map $T: V \...
user38140
6
votes
2
answers
2k
views
What is known about the spectrum of a Cauchy matrix?
Math people:
A Cauchy matrix is an $m$-by-$n$ matrix $A$ whose elements have the form
$a_{i,j} = \frac{1}{x_i-y_j}$, with $x_i \neq y_j$ for all $(i, j)$, and the $x_i$'s and $y_i$'s belong to a ...
- 523
2
votes
0
answers
393
views
A conjecture about vector space (repost from math.SE)
This post is copied from math.SE in the following link:
https://math.stackexchange.com/questions/456398/a-conjecture-about-vector-space
I have posted the question two days ago, but receive no answer ...
- 29
4
votes
3
answers
392
views
Algebraically Independent Numbers and Affine Linear Maps
Suppose I have two lists $\alpha_1,\dots,\alpha_k$ and $\beta_1,\dots,\beta_k$ of real numbers such that all $2k$ numbers are mutually algebraically-independent over the rationals. For each $i \in \{1,...
- 143
3
votes
1
answer
908
views
numerical range of a column-zero-sum matrix
I am trying to produce an example of a (necessarily non-normal) matrix that has only eigenvalues with positive real part, but whose numerical range contains elements with strictly negative real part. ...
- 3,328
4
votes
3
answers
3k
views
A nice necessary and sufficient condition on positive semi-definiteness of a matrix with a special structure
Let
$$
A =
\begin{pmatrix}
\sum_{j\ne 1}a_{1j} & -a_{12} & \cdots & -a_{1n}\\
-a_{21} & \sum_{j\ne 2}a_{2j} & \cdots & -a_{2n}\\
\vdots & \vdots & \ddots & \...
0
votes
1
answer
256
views
Name for a Specific Type of Non-Symmetric Bilinear Form
Let $V$ be a finite dimensional vector space, with some choice of basis $\{e_i\}_{i \in I}$. With respect to an idempotent bijection $B:I \to I$, define a bilinear form by
$$
g = \sum_{i=1}^N \lambda_{...
6
votes
3
answers
575
views
Reference for partial Hadamard matrices
Definition. An $m\times n$ matrix is said to be a partial Hadamard matrix (let's say PHM) if its entries are chosen from $\lbrace -1, 1 \rbrace$ such that the dot product of each pair of row vectors ...
- 1,985
7
votes
1
answer
512
views
Are dual spaces barreled?
Let $X$ denote a topological affine space (with no additional assumptions). Let $X^*$ denote its dual space of continuous affine functionals, equipped with the weak-$*$ topology. It is easy to see ...
- 8,392
2
votes
2
answers
259
views
Is my use of the eigendecomposition correct here?
I'm exploring different techniques to efficiently solve some matrix equations. My situation is that I have a matrix $\textbf{H} = \textbf{J}^T \textbf{J}$, where $\textbf{J}$ is a matrix with no ...
- 123
1
vote
0
answers
1k
views
Do skew symmetric matrices ever naturally represent linear transformations?
I am used to thinking of matrices in terms of linear transformations, but it occurred to me that skew-symmetric matrices are a potential counterexample.
I can think of at least two examples in which ...
- 7,199
2
votes
1
answer
117
views
Iterated Reduced Tensor Power of Graded Vector spaces
This might be inappropriate for the MO-level. If so I'll delete it...
Suppose $V$ is a $\mathbb{Z}$-graded vector space and
$\overline{T}(V):=V \oplus V\otimes V \oplus \otimes^3 V \ldots$
is the '...
- 624
4
votes
2
answers
349
views
A certain type of constrained Rayleigh-Ritz ratio
Let $\mathbf{A_1}$ and $\mathbf{A_2}$ be two hermitian matrices. Consider the problem
\begin{align}
\max_{\mathbf{u}^H\mathbf{u}=1}~\mathbf{u}^H\mathbf{A}_1\mathbf{u} \\\
\mathbf{u}^H\mathbf{A}_2\...
- 1,371
2
votes
1
answer
275
views
Possible pathological properties of positive definite matrix
Suppose $A$ is a positive definite matrix such that$$ I \preceq A \preceq 1.01I.$$ Is it possible that $\sum\limits_{i=1}^n A_{1i}$ can be arbitrarily large?
- 55
1
vote
1
answer
172
views
Estimating the relation between the covariance of a vector and a monotone function of the same vector
Let $\boldsymbol{X}\in\mathbb{R}^n$ be a random variable with positive entries ($X_i\geq a>0$). I want to characterize the relation between the second moment matrix $\boldsymbol{M}$, defined as
$$ ...
- 137
2
votes
1
answer
442
views
Positive definite to nonnegative
A simple question that I was pondering on while examining some algorithms that work similarly for positive definite and nonnegative matrices.
Let $\mathcal{H}$ be the space of (let's say for now $2\...
- 19.4k
0
votes
3
answers
1k
views
rank of outer product
I would like to ask if you may know how to prove this claim or any theorem related:
Given 9 points (x,y,z) lie on unit sphere in 3 dimensional space such that any 4 points are not on the same plane. ...
- 3
3
votes
2
answers
2k
views
Invariants of Matrix Reordering
are there any invariants of matrices, that are not affected by row- and/or column permutations?
To me it seems that the sequence of singular values could be such an invariant; am I right, resp. are ...
- 12.7k