All Questions
6,292 questions
15
votes
2
answers
7k
views
Efficient rank-two updates of an eigenvalue decomposition (or more generally SVD)
Let $A$ be a symmetric matrix with eigenvalue decomposition $UDU^T$. Golub, et al.1 and Bunch, et al.2 have shown that given such an $A$, the eigenvalue decomposition of $A+\rho xx^t$ may be computed ...
- 947
5
votes
0
answers
392
views
Preconditioner for finding the smallest eigenpairs of a large, but structured, matrix
I'm trying to find the eigenvector corresponding to the second smallest eigenvalue of a large $(4,000,000 \times 4,000,000)$ matrix $L$. $L$ is a graph Laplacian, with the following structure: $L = D -...
- 500
2
votes
0
answers
212
views
Compute the discriminant for reductive groups
Consider $G=GL_{2}$ and $F=k((\pi))$, and a diagonal matrix $t=\left(\begin{array}{cc}a&0\\0&b\end{array}\right)$.
The characteristic polynomial of $t$ is $X^{2}-(a+b)X+ab$, and the ...
- 3,472
2
votes
3
answers
755
views
On matrices in linear forms with vanishing determinant
This is a cross-post from my original question at math.se. I decided to post here because it seems more difficult than I originally thought.
Let $R=\mathbb C[x_1,\ldots,x_r]$ be a polynomial ring. ...
- 4,902
3
votes
2
answers
643
views
On a determinant inequality of positive definite matrices
Assume that $B$ and $A$ are two positive definite matrices. Take $B^*$ a block diagonal matrix with block $B_{11}$ and $B_{22}$ of $B$. This means the following:
$$
B=\left[\begin{array}{ll}
B_{11}&...
- 752
0
votes
1
answer
157
views
singular values function
Let $\mathbf{F}\in\mathbb{C}^{M\times M}$ and $\mathbf{D} = \operatorname{diag}(\mathbf{d})$ where $\mathbf{d}\in\mathbb{R}^M$. By SVD, $\mathbf{F}\mathbf{D}\mathbf{F}^H=\mathbf{U}\mathbf{S}\mathbf{U}^...
- 9
5
votes
2
answers
1k
views
How to calculate the determinant bundle
Maybe, this is a problem of linear algebra. But I do not know how to calculate it. Let $E$ be a vector bundle of rank $2$ over an algebraic surface. If $H=S^{2n}E\bigotimes (\operatorname{det} E)^{\...
- 713
4
votes
1
answer
609
views
Linear Complex Structure and Kahler Angles
I am trying to read Donaldson's paper on symplectic submanifolds
Link
and am getting a bit stuck on some simple linear algebra at the beginning. On the fourth page of the paper (p. 669) there is the ...
- 61
0
votes
2
answers
853
views
Kernel of $AB$ if $[A,B]=0$ and $AB\neq0$? [closed]
I have found similar results here and mathematics stack exchange but they all imposed specific conditions that don't suit this problem in particular. The problem is as follows.
Let A,B be square $n\...
- 39
26
votes
1
answer
3k
views
Linear Algebra without Choice
We consider the field of "usual" linear algebra.
Q. Which aspects of it can be carried out without the Axiom of Choice?
Q. Do interesting "exotic" phenomena appear in presence of (some instance of) ...
- 23.4k
2
votes
2
answers
509
views
Banach algebra of BV functions
I would like to find a reference for the proof that functions of bounded variation make a Banach algebra. Same question for $BV\cap L^\infty$.
- 16.2k
2
votes
1
answer
256
views
Find the transformation $P$ that minimizes the following:
$$\displaystyle\min_{\mathbf{P}} \text{trace}(\mathbf{APP^HA^H}) \quad{} \text{subject to} \quad{} \text{trace}(\mathbf{(I-P)(I-P)^H})=\alpha, \alpha \geq0$$ Can also be rewritten as $$\displaystyle\...
- 53
1
vote
0
answers
269
views
M-matrix with nonconstant entries properties
I have a matrix $J(x)$ with $J_{ij}(x)=f_{ij}(x)$ where vector $x$ is $x=x_1, x_2, ..., x_m$. I have shown that $J(x)$ is an M-matrix for all $x$. There is known review paper by Plemmons (1977) of 40 ...
- 11
0
votes
0
answers
115
views
a very elementary question on the conjugated matrices
Let $A$ and $B$ two matrices in $GL_{n}(K[[\pi]])$, regular semisimple on $GL_{n}(K((\pi)))$, with $K$ an algebraically closed field of characteristic zero .
We suppose that they have the same ...
- 3,472
0
votes
1
answer
2k
views
eigen-decomposition solution? is it unique?
Assume an N*N covariance matrix (Q) which is a positive definite matrix. The decoder X is assumed to be N*s, where s<=N. X is calculated to be s eigenvectors corresponding to s minimum eigenvalues. ...
- 1
3
votes
0
answers
466
views
An optimization problem over real symmetric matrices
Given an $n\times s$ matrix $P$ of positive real numbers and $T\geq n$, find (either by a formula or an algorithm) the real symmetric $n\times n$ Z-matrix $A$ which maximizes $\min\limits_{1\leq i<...
- 767
13
votes
3
answers
2k
views
"Values" of divergent integrals
Are there existing theories of integration in which $I_0 = \int_0^{\infty} dx$ and $I_1 = \int_0^{\infty} x \ dx$ are well-defined infinite elements in a non-archimedean extension of the reals? I can ...
- 19.7k
2
votes
1
answer
362
views
Matrix power problem
$J$ is a symmetric matrix (built from 6j symbols...it's always knot theory in disguise when I ask :-), $D$ a diagonal matrix, and $B=DJ$. $S$ is a diagonal sign matrix (entries all $+1$ or $-1$). $I$ ...
- 4,829
2
votes
0
answers
1k
views
Linear Algebra Text Book [closed]
In our department we do not like our current linear algebra book and so we would want to find a better book. This is for the first course in linear algebra and the title of the course is
Elementary ...
2
votes
1
answer
127
views
Existence of an "Orthogonalizing" Operator
I was wondering if it was possible to prove existence of a unitary operator $A$ such that:
$\langle Au,u\rangle=0$ for all $u$.
In 2-dimensions it clearly is (just a 90 degrees rotation) and similarly ...
- 123
0
votes
1
answer
140
views
QR alogrithm for eigenvalue problem [closed]
Considering pure QR algorithm (without shifts and preliminary tridiagonal reduction) are there sufficient conditions for algorithm to converge to quasi-diagonal form?
For the the following matrix
$$
...
1
vote
1
answer
172
views
Linear combinations of basic cubes on a torus board
Consider an $n \times n \times n \times\dots\times n$ torus board of total size $n^k$ with $n > 4$ either even or odd.
Consider the basic cube of size $1 \times 1 \times \dots \times 1$ at a ...
- 13.9k
0
votes
1
answer
274
views
when $g^*$ is invariant under $Ad(G)$?
Let $G$ be a Lie Group and $\mathfrak{g}$ be its lie algebra.
Let $\mathfrak{g}$ is semisimple or reductive lie algebra, then prove that $\mathfrak{g}^*$ (dual of $\mathfrak{g}$)is invariant under $...
user21574
16
votes
1
answer
3k
views
A property that forces the NORM to be induced by an INNER PRODUCT
Let $(E, \|\cdot\|)$ be a real normed vector space such that for any $a,b\in E$,
$$ \|x +y\|^2 + \|x-y\|^2 \geq 4 \|x\|\cdot \|y\| $$
I want to show that the norm is induced by an inner product. Any ...
user36763
13
votes
2
answers
4k
views
Writing a matrix as a sum of two invertible matrices
Let $n\geq 2$. Is it true that any $n\times n$ matrix with entries from a given ring (with identity) can be written as a sum of two invertible matrices with entries from the same ring ?
user39321
44
votes
2
answers
2k
views
Is this lemma in elementary linear algebra new?
Is anyone familiar with the following, or anything close to it?
Lemma. Suppose $A$, $B$ are nonzero finite-dimensional vector spaces
over an infinite field $k$, and $V$ a subspace of $A\otimes_k B$
...
- 595
0
votes
1
answer
606
views
Number of Minimal left ideals in the full matrix ring over a finite commutative local ring
Inspired with another QUESTION I would like to know the number of minimal left ideals of $M_n(R)$ in terms of $n$ and $R$ where $R$ is a finite local commutative ring with identity ?
user39204
4
votes
0
answers
2k
views
eigenvalues of a symmetric tridiagonal matrix with zero diagonals
I was investigating a problem and came up with the following symmetric tridiagonal matrix (with zero diagonal elements):
$$
\left(\begin{array}{cccccc} 0 & a & 0 & \ldots & 0 \\ a &...
- 103
8
votes
1
answer
586
views
Main problems on lattice-basis reduction algorithms (such as LLL)?
What are the main open problems on lattice-basis reduction algorithms (such as LLL)? I am looking for problems satisfying the following two conditions:
(a) their solution would likely be of some ...
- 20.2k
2
votes
1
answer
228
views
Subgroup of $SL(n,\mathbb{R})$ with positive entries
Let $SL(n, \mathbb{R})$ be the group of $n \times n$ invertible matrices of determinant $1$ in real numbers. Let $G:=SL(n, \mathbb{R}_{\geq 0})$ be its subgroup $\{M \in SL(n,\mathbb{R}) \mid M, M^{-...
- 3,472
1
vote
1
answer
517
views
Estimate on the real and imaginary parts of eigenvalues
Let $A$ be a matrix. If $A$ is "almost" equal to $A^*$, it follows from an argument of continuity that the eigenvalues of $A$ are "almost" real. Same argument can be made for $A$ "almost" $-A^*$, in ...
- 113
0
votes
1
answer
328
views
Number of minimal left ideals
Is there any way to compute the number of minimal left ideals of $M_n(K)$, the full $n\times n$ matrix ring with entries in the field $K$ ?
user39204
4
votes
1
answer
1k
views
Can two unitary similar real matrix be orthogonal similar
Can two unitary similar real matrices be orthogonal similar?
suppose $A=U^tBU$ where $U$ is unitary, does there always exists a real orthogonal matrix $O$, such that $A=O^tBO$ ?
user39380
5
votes
0
answers
112
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
507
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 $V$. ...
- 7,249
0
votes
1
answer
242
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
813
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 ...
- 3,932
0
votes
1
answer
106
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
165
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
784
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
706
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
825
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
$$
||z_1||=\ldots = |...
8
votes
2
answers
425
views
Dimension of commutative subalgebras of a central simple algebra
let $k$ be a field, and let $A$ be a central simple $k$-algebra over $k$.
What is the maximal dimension of a commutative $k$-subalgebra of $A$?
If $A=M_r(D)$, where $D$ is a central division $k$-...
- 81
6
votes
2
answers
994
views
Minkowski successive minima inequality for a lattice base?
Let $\Lambda$ be a lattice of $\mathbb{R}^n$, and $\lambda_i$ be the radius of the smallest ball containing $i$ linearly independent lattice vectors.
The Minkowski successive minima inequality says ...
- 403
1
vote
0
answers
783
views
Determine lattice basis from given lattice points
I'm working on the Shortest Lattice Vector Problem (SVP) for a paper that I'm currently writing. I wish to verify whether a particular structural, namely the building block property ( refer to the ...
- 11
4
votes
1
answer
429
views
Finding all local maximum points of a function?
Let ${\boldsymbol \theta}=(\theta_1,\theta_2,\ldots,\theta_n) \in{\mathbb T}^n$ and $P:{\mathbb T}^n\rightarrow {\mathbb R}$ be a function defined on $n$-torus as
$$
P({\boldsymbol \theta}) = \sum_{i&...
1
vote
1
answer
177
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 ...
- 1,547
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
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