All Questions
6,176 questions
1
vote
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Extending linear operators to multi-linear ones
Suppose we are given a linear operator $L$ on a Banach space $X$. Is there any way to extend $L$ to a multi-linear operator $\mathcal{L}$ in such a way that
$$\mathcal{L}(x_1, x_2^*, \ldots, x_n^*) = ...
- 73.2k
8
votes
1
answer
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Is there an elementary way to show the triangular inequality for this expression ?
Consider the space $X$ of all scalar products on $\mathbb{R}^n$. For a scalar product $s$ and a base $B:=b_1\ldots,b_n$ let $M_{s,B}$ denote the matrix, whose $(i,j)$-th entry is $(s(b_i,b_j))$ . ...
- 28.6k
18
votes
1
answer
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Commuting unitaries
Is the following true:
For every unit vectors $x_1,..., x_n$, $y_1,..., y_n$ in $\mathbb{C}^k$
there exist a Hilbert space $H$, unitary operators $U_1,...,U_n$ and $V_1,...,V_n$ in $B(H)$ and unit ...
- 4,705
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votes
1
answer
322
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Sparse Principal Components Analysis: Any practical examples with fixed rank correlation matrix?
Consider the problem of sparse principal component analysis:
$$\max_{||{\bf x}||_0=k,||{\bf x}||_2=1} {\bf x}^T{\bf A}{\bf x}$$
where a $k$-sparse $n$-dim. unit vector that "maximizes variance" is to ...
- 96.4k
-1
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1
answer
185
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eigenvalues of $I\otimes B\otimes C + A\otimes I \otimes C + A\otimes B \otimes I $
Let $A$, $B$ and $C$ be symmetric matrices.
What can we say about eigenvalues of $I\otimes B\otimes C + A\otimes I \otimes C + A\otimes B \otimes I $?
- 56.6k
2
votes
1
answer
205
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Do unitary bijections act invariantly on irreducible representations?
Let $\mathcal{A}$ be a $C^*$ algebra. Let $(\pi, \mathcal{H})$ be a faithful, irreducible, unitary, Hilbert space representation of $\mathcal{A}$; i.e., $\pi:\mathcal{A}\rightarrow\mathcal{B}(\mathcal{...
- 25.5k
3
votes
1
answer
376
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On some type of conjugate of elements of SL(n,Z)
Let $A\in SL(n,\mathbb{Z})$ and $B\in\mathcal{M}_n(\mathbb{Z})$ s.t. $\det(B)\ne 0$. Is it possible to find a power of $B^{-1}AB$ in $SL(n,\mathbb{Z})$?
- 1,832
4
votes
1
answer
3k
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Cauchy-like inequality for Kronecker (tensor) product
General question first: upper/lower bound a sum of Kronecker products by its components. More specifically,
how is $$
\Vert\sum_{\alpha}S_{\alpha}\otimes B_{\alpha}\Vert$$
bounded by the operator ...
- 731
6
votes
3
answers
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Is there a notation for the symmetric / antisymmetric subspaces of a tensor power that distinguishes them from the symmetric / exterior power?
Let $V$ be a finite-dimensional vector space over a field $k$, say of characteristic $0$. The symmetric group $S_n$ acts on the tensor power $V^{\otimes n}$ in the obvious way, and this action defines ...
CommunityBot
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votes
2
answers
308
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Analogue of an orthogonal subspace in a noneuclidian normed space
This question is related to https://mathoverflow.net/questions/50600/an-existence-question-on-linear-map. If the answer to this question is yes, it would solve the abovementioned other MO question.
...
CommunityBot
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3
votes
3
answers
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a "reverse Hadamard inequality"
Is there an inequality of the form $|\det(A)| \geq F(v_1, \ldots, v_n)$ for a real $n\times n$-matrix $A$ with columns $v_i$, $F \geq 0$?
- 52.3k
3
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0
answers
681
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How to bound the second largest eigenvalue of a transition matrix of a non-irreducible Markov chain?
I have found several bounds (e.g., Cheeger, Poincare) for the case that the Markov chain is irreducible and reversible, however my Markov chain has one absorbing state. Any bound would be helpful, but ...
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1
answer
464
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eigenvalues of A⊕B
Let $A_{n\times n}=(a_{ij}),B_{n\times n}=(b_{ij}) \in M_{n}(\mathbb{R})$, where $a_{ij},b_{ij} \in \lbrace 0,1\rbrace$. Boolean sum of $A,B$ denoted by $(A \oplus B)_{n\times n}=(a_{ij}\oplus b_{ij})$...
- 19.6k
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4
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Prove: if a1,...,an are uniformly distributed unit vectors, then a1*a1'+...+an*an'=n/2*I
Hello everyone,
I have a very interesting question on orthogonal projection matrices. Intuitively it is quite straightforward to understand. But for me it is not easy to prove.
In $R^2$ space, $a_i$,...
- 16.9k
4
votes
1
answer
1k
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dominant eigenvector
Hi, everyone! Is there any efficient way to simplify the following tensor product
$X \otimes X + X^T \otimes X^T$, where $X$ is a square $n \times n$ matrix.
My goal is to efficiently compute the ...
- 20.2k
1
vote
1
answer
149
views
Question on a relation between minors of a particular kind of matrix
Hi!
Perhaps it is an easy question but i don't figure out how to prove it.
Let $a_1,...,a_{2m+2}\in\mathbb{C}$ and for $1\leq i\leq 2m+2$ and $j\leq [\frac{2m+2-i}{2}]$ (with $[a]$ i mean the integer ...
CommunityBot
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votes
3
answers
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Which vector spaces are duals ?
Every finite-dimensional vector space is isomorphic to its dual.
However for an infinite-dimensional vector space $E$ over a field $K$ this is always false since its dual $E^\ast$ is a vector space ...
CommunityBot
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votes
1
answer
1k
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Whether the system of matrix equations is always solvable
In recent days, I learned a linear algebra problem from one of my friends.
It can be stated as follows.
Given four matrices $A,B,C,D$, find three matrices $E,G,F$, not simultaneously zero, such that ...
- 4,705
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votes
1
answer
2k
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Solving 5 eqns with 6 unknowns in a 2x3 contingency matrix, is there a unique solution? [closed]
Background
I have the following equations:
$$a+b+c=6$$
$$d+e+f=15$$
$$a+d=5$$
$$b+e=7$$
$$c+f=9$$
This is a 2x3 matrix $[a b c, d e f]$ where the marginal totals are fixed. In addition, all of the ...
CommunityBot
- 1
16
votes
3
answers
3k
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A riddle about zeros, ones and minus-ones
I was asked this years ago, but I don't remember by whom, and have never managed to solve it.
Consider the $2^n \times n$ matrix of all vectors in {-1,1}$^n$.
Someone comes and maliciously replaces ...
- 11
4
votes
1
answer
3k
views
SVD complexity for structured sparse matrices
Hello,
For an $n \times n$ real matrix, the SVD (Singular Value Decomposition) algorithm is $O(n^3)$.
I have large matrices (say $10,000 \times 10,000$) that only have elements on few diagonals, i.e. ...
- 96.4k
13
votes
3
answers
2k
views
Relationship between determinants.
Given an orthogonal matrix $O$ with dimensions $4n \times 4n$ and $\det O = -1$, how to prove that
$\det[O_{11} - O_{22} + i (O_{12} + O_{21})] = 0$?
Here $O$ is a block matrix $[[O_{11}, O_{12}], [...
CommunityBot
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5
votes
2
answers
2k
views
Lower Bound on the Cost of Solving Linear System
The cost of solving a linear system ("exactly") with Gauss Elimination and other similar methods with a few right hand side and where the matrix has no structure is $\mathcal{O}(N^3)$ where $N$ is the ...
CommunityBot
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1
vote
1
answer
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Pseudoinverse of columns of a matrix
First, some background:
I'm working on an implementation in C# of Lemke's algorithm (for solving linear complementarity problems) based on this Matlab implementation: http://ftp.cs.wisc.edu/math-prog/...
- 3,934
6
votes
2
answers
364
views
Algebraic characterization of transitive spaces of matrices
Fix an integer $d \ge 2$ and let $M_d$ be the space of real $d \times d$ matrices. Let $E$ be a vector subspace of $M_d$. We say that $E$ is transitive if $E \cdot \mathbb{R}^d_* = \mathbb{R}^d$, ...
- 2,479
2
votes
1
answer
3k
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Is it possible to decompose a symmetric, positive definite matrix in this way?
Let $\Sigma$ be a symmetric positive definite matrix. Then the Cholesky decomposition gives us $\Sigma=LL'$ where $L$ is lower triangular and unique.
Under what conditions (if any) does there exist ...
- 96.4k
1
vote
0
answers
466
views
Bounding point-wise maximum of the absolute difference of two convex functions
Let $\Delta: R \times R \rightarrow R_{+}$ be a positive and convex function (convex in, say, both the arguments) called the loss function.
Let $x \in R^d$. Moreover, let $H_1,...,H_r$ be sets of ...
- 11
3
votes
2
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428
views
finding an element of a vector subspace contained in the first orthant
Given a matrix $M$, I want to find a nontrivial vector in the kernel of $M$ that also lies in the first orthant, if such a vector exists. That is, I want to simultaneously solve
$$Mx = 0$$
$$x \geq 0$...
- 44.7k
7
votes
2
answers
1k
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Row reduction of sparse matrices
Let $p$ be prime (of size roughly $100$, say). Suppose that $M$ is a matrix with coefficients in $\mathbf{F}_p$ with roughly $An$ rows and $n$ columns, where $A>1$ is some fixed small constant. ...
CommunityBot
- 1
12
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5
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2k
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Is this formulation of the Singular Value Decomposition standard?
In customary formulations of the Singular Value Decomposition or SVD that I have seen,
(e.g., Wikipedia or Gil Strang's textbooks) it is always stated in terms of writing an
$m \times n$ matrix $M$ (...
- 52.3k
5
votes
1
answer
263
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Modeling free Lie algebras with matrix algebras
I am approximating some algebraic expressions of operators from a free Lie algebra. It is possible but messy to collect all independent operator objects of a given degree (same as grading?) that ...
- 2,804
2
votes
1
answer
331
views
Symmetric polynomials preserving $-1,1$ matrices
If $A$ is an $n\times n$ integer matrix, then trivially $S=A+A^t$ and $P = AA^t$
where $t$ is ``transpose", are both symmetric.
Assume that $A$ is also a "$\lbrace -1,1 \rbrace$" matrix, i.e., the ...
- 13.5k
3
votes
4
answers
6k
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Applied linear algebra textbook? [closed]
I have a copy of Linear Algebra Done Right, which I worked through years ago in college. I have been using that book to refresh my knowledge, but it does not have an applied or computational aspect ...
- 15.3k
1
vote
1
answer
304
views
How do maximum norms relatively change in Euclidean translations
Let $Q$ be the cube $[-1,1]^{3}$ and $\pi$ be a plane in $\mathbb{R}^{3}$
that contains the origin but doesn't contain any vertex of $Q$. Suppose that $A$ is an invertible
linear transformation from $\...
- 11
3
votes
0
answers
528
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A question about the generalized Lidskii-Wielandt inequality for matrices proved by Thompson and Freede
In 1971, Thomson and Freede generalized the Lidskii-Wielandt inequalites as follows (version for singular values)
Let $A$, $B$ be $n\times n$ Hermitian matrices. Suppose $\alpha_1\geq \alpha_2 \geq \...
- 52.3k
1
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1
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479
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Is exp(rA) = (exp(A))^r for real r and A in a Banach space?
Is $e^{(rA)} = (e^{A})^r$ when $r \in \mathbb{R}$ and $A$ is an element of a Banach algebra?
Clearly if $n$ is an integer, then
$e^{(nA)} = e^{A+A \cdots +A} = e^{A}e^{A}\cdots e^{A} = (e^{A})^n$,
...
- 7,237
1
vote
3
answers
2k
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Principal curvatures and curvature directions [closed]
Last week I considered again principal curvature (pc) and principal curvature directions (pcd) of a, for the sake of simplicity, 2-manifold embedded in 3-space. In this simple case, the pc and pcd of ...
- 2,437
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0
answers
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Inverse of a matrix with binomial coefficients
Let $a(n,k)=(-1)^k {{2n-k}\choose k}$ for $0 \le k \le n$ and $a(n,k)=0$ else. Then it is known (cf. OEIS A005439 and A098435) that the first column of the inverse matrix of $(a(i,j))_{i,j\ge0}$ is ...
- 5,622
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2
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2k
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a matrix similarity problem.
I'd like to know whether the following statement is true or not.
Let $T_1, T_2\in \mathbb{C}^{n\times n}$ be upper triangular matrices. If there exists a nonsingular matrix $P$ such that $T_1=PT_2P^{-...
- 4,736
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2
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1k
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Assessing measurement accuracy and precision
I have been asked to assess the accuracy and precision of a new measurement method (Let's call it method B). This new method is compared to an existing one (A) that has its own specifications in ...
CommunityBot
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1
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0
answers
2k
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Tensor Products and Intersections
Given two algebras $A$ and $B$, and two ideals $I, J \subseteq B$ with non-empty intersection, is it true that
$$
(A \otimes I) \cap (A \otimes J) = A \otimes (I \cap J)?
$$
(Where both sides of the ...
- 1,472
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
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1
answer
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Eliminating redundant linear constraints? [closed]
I have an NxN matrix of linear constraints that is not of full rank. In other words, some of the constraints are linear combinations of other constraints. The "standard" linear algebra tools (...
- 1,743
2
votes
2
answers
492
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on existence of matrices X, Y s.t. XAY is diagonal over non-commutative ring
Given $A\in Mat_{n\times n}(R)$ where $R$ is a non-commutative associative ring are there exist any (non-zero) matrices $X, Y\in Mat_{n\times n}(R)$ such that $XAY=diag(a_1, \ldots , a_n)$ for some $...
CommunityBot
- 1
1
vote
0
answers
265
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"Lift and project" procedure for matrices
Definition. Let us call $n\times n$ matrix with non-negative entries good if sum of every row and column is equal to $1/n$.
Suppose we have a good matrix $A$. Let us consider the following strange "...
- 1,791
5
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0
answers
391
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An operator-norm version of Siegel's Lemma
Is there a kind of Siegel's Lemma saying that if $M$ is a ``small-height'' integer matrix, then there is a "small-height" vector $x$ with $\|Mx\|=\|M\|\|x\|$? (Here $\|Mx\|$ and $\|x\|$ denote the ...
- 23k
9
votes
2
answers
2k
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Basis for the Algebraic numbers over the rationals
Is there an explicit basis for the algebraic numbers as a vector space over the rationals?
- 269
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0
answers
1k
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Can we pass to the limit in Poincaré-Jaynes-Bretthorst interpolation and deconvolution?
In Science and Hypothesis, chapter XI, The calculus of probabilities, Henri Poincaré deals informally with the fundamental problem of interpolation. He concludes (see http://ia600308.us.archive.org/21/...
2
votes
1
answer
406
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Are these systems of linear equations always solvable
Let $X$ be a (finite) simplicial complex and let $f$ be a map from the set of its $n$-Simplices to a abelian group $A$, with the property, that every cycle maps to $0$ (extending $f$ linearly).
Let $...
- 73.2k
0
votes
1
answer
225
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Codimension of non-common condition is 2?
If we have n homogeneous polynomials (over algebraically closed field) $f_1\ldots , f_n$ on variables $x_0, \ldots , x_n$
$$
f_i(x_0, \ldots , x_n) = \sum_{j_0,\ldots , j_n} a_{i, j_0, \ldots , j_n} ...
- 42.9k