Questions tagged [linear-algebra]
Questions about the properties of vector spaces and linear transformations, including linear systems in general.
5,677
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Eigendecomposition of analytic Hermitian matrix-valued functions of several variables
If $A(t)$ is an analytic, Hermitian matrix-valued function of a real variable $t$, then it is known that there are analytic functions $\lambda_i(t)$ and $x_i(t)$ corresponding to the eigenvalues and ...
- 585
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799
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Regular or elliptic elements in the multiplicative group of central division algebra
For an element $g$ of a connected reductive group $G$ over a field $F$,
$g$ is called $regular$ if the dimension of the centralizer of $g$ is equal to the rank of the algebraic group $G$,
$g$ is ...
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468
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Isomorphisms between topological vector spaces [closed]
Let $f : X \to Y$ be a continuous map of complete topological vector spaces. Suppose that $A \subseteq X$ and $B \subseteq Y$ are proper, dense linear subspaces, and that the restriction map $f : A \...
- 8,392
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Fields whose embeddings into the complex numbers are invariant under complex conjugation
Is there a general notion/description of fields $K$ such that the image of any embedding $K \hookrightarrow \mathbb{C}$ is invariant under complex conjugation, thus inducing an involution on $K$ which ...
- 41
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787
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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
4
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623
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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?
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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....
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How to efficiently compute the generalized cross product?
It's possible to extend the well known cross product between two vectors in $\mathbb{R}^3$ to $n-1$ vectors in $\mathbb{R}^n$.
Let $\vec{v_1}, \vec{v_2}, \dots, \vec{v}_{n-1} \in \mathbb{R}^n$ and $\...
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2
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Woodbury formula
I wonder - do you know of any example where the Woodbury formula (cf. http://en.wikipedia.org/wiki/Woodbury_matrix_identity) was crucially used to prove anything?
It might be a useful computational ...
- 6,872
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signs of eigenvalues of quadratic form
Let $A=(a_{ij})_{i,j=1}^n$ be a symmetric real matrix, $M_k:=det(a_{ij})_{1\leq i,j\leq k}$ be its minors and $M_k\ne 0$ for all $k$. Then signs of eigenvalues of $A$ are equal (up to some permutation)...
- 103k
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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 ...
- 25.3k
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Stability of Conjugate Gradient Method
When dealing with spd matrices of relatively low condition number, how likely is it (and is it easily provable) that the conjugate gradient method will always be able to find the solution without ...
- 221
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253
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Why do these polynomials split almost in the middle?
Start with a palindromic sequence of integers $(a_0, a_1, \ldots, a_{n+1})$, i.e. $a_j=a_{n+1-j}$, and put $a_j:=0$ for $j<0$ and $j>n+1$. You may readily guess that the choice of the binomial ...
- 13.2k
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Singular value decomposition of truncated discrete Fourier transform matrix
Let $\mathbf{F}$ be a discrete Fourier transform (DFT) matrix such that
\begin{align}
F_{m,n}=e^{-j2\pi(m-1)(n-1)/N},\quad m,n=1,\ldots,N.
\end{align}
What we can say about the singular value ...
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The upper bounds on rank $ 2 $ real matrices
Let $ A_{n}(F) $ be the collection of all skew-symmetric matrices over the field $ F $ ($\operatorname{char} F \neq 2 $). Let M be a subspace of $ A_{n}(F) $ such that all non zero elements have rank ...
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Is it possible to complete a basis for a free module over a finite-dimensional associative unital real algebra?
Let $\mathbb F$ be a finite-dimensional associative unital real algebra. Consider $V:=\mathbb F^n$ and let's say $p \in V$ is good if $xp=0$ only has $x=0$ as solution.
Question: If $p_1$ is good, ...
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matrix congruence and smith normal form
Fixed $n \geq 2$ and consider $A,B \in GL(n,\mathbb{Z}).$ We know that we have the Smith normal form. One can find $U, V \in SL(n,\mathbb{Z})$ such that $A=UDV.$ So as $B$. The Smith normal form is ...
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Weierstrass approximation theorem proof by Gram-Schmidt orthogonal polynomials?
I am familiar with Bernstein proof of Weierstrass Theorem. However, I am curious whether it can be proven by using Gram-Schmidt orthogonalization of $x^n, n=0,\dots,N$ monomials? As $N\rightarrow\...
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Spectrum of sum of weighted Wishart matrices
This is a repost from mathstackexchange, as I think asking this question is more appropriate here.
Coming from statistical physics, I am interested in the (real) spectrum of the following sum, and ...
- 41
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2
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Example of an inverse system which suddenly "jumps" in size in a specific "controlled" way?
I'm looking for an inverse system $(X_\alpha)_{\alpha < \omega_1}$ of vector spaces (EDIT: over a finite field) such that, for some $\lambda \geq 2$ with $\lambda < \lambda^{\omega_1}$ (I ...
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Is the Loewner maximum uniquely defined?
Given 2 (symmetric) PSD matrices $A,B$, is the following set $S_{A,B}$ non-empty?
$$ S_{A,B} = \{ C: C\succeq A, C\succeq B, \text{ and }\forall D, D\succeq A, D\succeq B \implies D\succeq C \} $$
If ...
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When is rank-1 perturbation to a positive operator still positive?
Let $A : \mathcal{H} \to \mathcal{H}$ and $B : \mathcal{H} \to \mathcal{H}$ be trace-class (hence compact) Hermitian operators on a separable Hilbert space. Assume that $A$ is strictly positive and ...
- 650
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Problem arising in metrizability of connections: Simultaneously skewsymmetrizing matrices
Fact: Let $U$ and $V$ be two $ n \times n$ matrices with determinant $ 1.$ Assume that $S_1,S_2,....S_m$ are linearly independent $n \times n$ matrices such that $U^{-1}S_iU$ and $V^{-1}S_iV$ are ...
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Commensurator of a subgroup of matrices
Let $k$ be a totally real number field and let $\mathcal{O}_k$ denote its ring of integers. If $H$ is a subgroup of $\text{GL}(n, \mathbb{R})$ let denote with $H(k)$ and $H(\mathcal{O}_k)$ the ...
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Linear maps preserved by algebra automorphisms
Let $A$ be a finite dimensional , associative, unital $F$-algebra, where $F$ is a field.
Let $s_A:A\to F$ be an $F$-linear map.
Now consider an arbitrary field extension $K/F$, and define $s_{A\...
- 1,661
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401
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Lipschitz property of matrix function only depending on singular values
Let $f$ be a function from $\mathbb{R}^{n\times n}$ to $\mathbb{R}$ such that there exists another symmetric function $g$ (invariant under permutation of coordinates) from $\mathbb{R}^{n}$ to $\mathbb{...
- 1,475
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Convex Hull of Outer Products of (Normalised) Nonnegative Vectors
If I define $\mathcal{A} = \{ xx^T : x \in \mathbb{R}^d, \| x \|_2 \leqslant 1 \}$, then (assuming I recall correctly) it is known that the convex hull of $\mathcal{A}$ is given by
\begin{align}
\...
- 706
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Which groups can be reconstructed from a single invariant subspace?
Let $G\subseteq\mathrm{Perm}(\Bbb R^n)$ be a matrix group consisting of permutation matrices acting on $\Bbb R^n$. Let $U\subseteq\Bbb R^n$ be an irreducible invariant subspace w.r.t. $G$. Now, define ...
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Is there any Menelaus-type theorem for polynomials?
Consider $n+3$ polynomials of degree $n$, say $P_1(x) , ... P_{n+3}(x)$.
In addition, consider that there are distinct (it is not necessary that all of them be distinct) numbers $x_{ij}$ for $1 \...
- 550
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Determinant of structurally symmetric $n$-banded matrix?
Is there a formula to compute the determinant of a structurally symmetric $n$-banded matrix? I am specifically interested in the 5-banded matrix:
$$ \left[\begin{matrix}
c_{0} & s_{0} & 0 &...
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2
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458
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The Aleksandrov-Fenchel inequality of mixed discriminants for Hermitian matrices
Suppose $A,A_1,\ldots,A_{n-2}$ (resp. $B$) are (resp. is) real positive-definite (resp. arbitrary) symmetric $n\times n$ matrices and denote by $D(\cdot,\ldots,\cdot)$ the mixed discriminant. We have ...
- 483
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Distribution of regular elements in a disconnected algebraic group
Let $k$ be an arbitrary field (in my case $k = \mathbb Q_p$), and $G\subset \mathrm{GL}(n)_{/k}$ a reductive group. Let $G^0$ be its identity connected component.
Suppose that $G^0(k)$ contains an ...
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Vector bundles on Grassmannians
Let $Gr(k,n)$ be the Grassmannian of $k$-dimensional vector subspaces $H^k$ of an $n$-dimensional vector space $V$.
Let us fix an $h$-dimensional vector subspace $\Gamma\subset V$ with $h\leq k$, and ...
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Continuous linear combination of continuously varying vectors?
Let ${\bf{e}}_1, {\bf{e}}_2, {\bf{e}}_3:[0,1]\rightarrow \mathbb{R}^3$ be continuous, $\mathbf{0}\neq \mathbf{v}\in \mathbb{R}^3$.
Suppose that the following condition (C) holds:
$$
\exists d>0: ...
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votes
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answer
400
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minimizing the product of rayleigh quotient
The problem is:
$$\min_{\alpha}\frac{\alpha^T A \alpha}{\alpha^T\alpha}\frac{ \alpha^T B \alpha}{\alpha^T\alpha}$$
where $A$ and $B$ are symmetric and positive definite matrix.
I think the explicit ...
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Size of connected components of a graph via its spectrum
I know that we can determine the number of connected components of a graph from the eigenvalues of its Laplacian matrix. My question is:
Is there a way to understand the size of each connected ...
- 4,969
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Why are the vectors with this special structure linearly independent with high probability?
Given $a_i\in\mathbb{R}^m$ for $i=1,\ldots,2m$ with independent and identically random entries of some continuous distribution. Every choice of $m$ vectors from $\{a_1\ldots,a_{2m}\}$ is linearly ...
- 271
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subset of hermitian matrices given by eigenvalues form a submanifold
Let $\mathcal{O}_\lambda$ be the set of hermitian $n+1 \times n+1$ matrices with Eigenvalues $\lambda = (\lambda_1, \dots, \lambda_{n+1})$.
and $\mathcal{O}^\mu$ the set of hermitian $n \times n$ ...
- 501
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$\text{GL}(n + k, \mathbb{R})$ is a principal $H$ bundle over the Grassmann manifold $G_n(\mathbb{R}^{n+k})$?
Let $H$ be the subgroup of $\text{GL}(n + k, \mathbb{R})$ consisting of matrices whose lower $n \times k$ block is empty; i.e. consisting of matrices of the form$$\begin{pmatrix} A & * \\ 0 & ...
- 85
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Affine hull of a set of non-negative matrices with fixed row-sums
Fix any non-negative matrix $M \in \mathbb{R}_{\geq 0}^{m \times n}$ that contains no zero-row and no zero-column.
Further, fix any positive vector $r \in \mathbb{R}_{> 0}^m$.
With $nz(M) := \{(i,j)...
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Matrix Elements of Real Representations
I asked this question over at Math.StackExchange and despite having had a bounty on it I did not receive an answer.
Suppose that $G$ is a finite group and we have a unitary irreducible representation ...
- 966
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2
answers
460
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Extremal eigenvalues & eigenvectors of skew-adjacency matrix
I am looking for ways to obtain the extremal eigenvalues and eigenvectors of the skew-adjacency matrix of a directed graph without diagonalizing it. The graphs I am interested in are not regular (but ...
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Let $A\in\operatorname{M}_n(F)$ be a matrix, how to prove $\bigcap_{X\in C(A)}C(X)=F[A]=\frac{F[x]}{(m_A(x))}$
I asked the following question on Math Stack Exchange, but no people reply. I know MO is more professional and it is for mathematicians to discuss research problems. Maybe this question is unsuitable ...
- 243
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668
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$\mathcal{H}$-polyhedron under a linear map
Let $P = \{ x \in \mathbb{R}^n \mid Ax \leq b \}$ be a (bounded) polyhedron for $A \in \mathbb{R}^{m \times n}$ and $b \in \mathbb{R}^m$, $n,m > 0$.
Moreover, let $M \colon \mathbb{R}^n \to \...
- 321
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564
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Smith Normal Form for block matrices over the integers
Are there any known results on the Smith Normal Form for block matrices over the integers?
In particular, I am interested in matrices of size $kr \times ks$ made of square blocks of size $k$ such that ...
- 41
4
votes
1
answer
188
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weak version of a Baez-Crans 2-vector space?
Baez and Crans defined a 2-vector space to be a category internal to the category of vector spaces (say over the reals). I am interested in categories that are equivalent to Baez-Crans vector spaces ...
- 1,670
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1
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161
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Sensitivity of the range of a matrix
The distance between two subspaces $\mathcal{U}$ and $\widetilde{\mathcal{U}}$ is classically defined as $d(\mathcal{U},\tilde{\mathcal{U}}):=\|P-\tilde{P}\|$, where $P$ and $\tilde{P}$ are orthogonal ...
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756
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What is a degenerate Legendre Transformation?
I am studying the Lagrangian and Hamiltonian description of some dynamical systems. The problem with this description of the particular kind of systems I am studying, is that the Legendre ...
- 245
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1
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474
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Isomorphism of matrix ring over ore domain
Let $R_1,R_2$ be (left and right) ore domains. Does $ Mat_n(R_1)\cong Mat_m(R_2)$ implie m=n and $q.f.(R_1)\cong q.f.(R_2)$?
An counter example, a proof or a reference is welcomed.
Thanks
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3
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Solving a quadratic matrix equation with fat matrix
I am trying to find an $n \times m$ fat (i.e., $m > n$) matrix $T$ that solves
$$T^T T = X$$
where $X$ is a given $m \times m$ symmetric, positive semidefinite matrix.
I saw this post, but ...
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