All Questions
414 questions
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Euclidean volumes under the matrix norm of one-parameter subsets of unit balls of $2 \times 2$ matrices
Prove that the Hilbert-Schmidt volume (normalized to equal 1 at $\varepsilon=1$) of the subset of the unit ball in operator norm (http://mathworld.wolfram.com/OperatorNorm.html) of the $2\times 2$ ...
- 723
1
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1
answer
187
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A combinatorial matrix reconstruction problem II
For a positive integer $n$, let an $n$-shuffle be a multiset
$S=[(S_i,d_i)|i=1,\ldots,n]$ of pairs $(S_i,d_i)$, where each
$S_i$ is a multiset of $n$ numbers containing the number $d_i$.
A realization ...
- 3,873
1
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0
answers
264
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Maximizing the ratio of largest eigenvalues
Let $K$ be a real stable matrix; more specifically,
$$
K=\left(\begin{array}{rrrrr}
0&1&0&\ldots&0\\
0&0&1&\ldots&0\\
\vdots&\vdots&\vdots&\ddots&\vdots\...
- 270
1
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1
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171
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transposing "unrimmed" permutations
Denote the set of $n\times n$ permutation matrices by $\mathfrak{S}_n$. The ordinary transpose preserves this group.
Given $P\in\mathfrak{S}_n$, construct the $n\times n$ matrix ${}^tP$ according to ...
- 43.2k
1
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267
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Construct special "joint SVD" from separate SVDs
Given two matrices, $A,B\in\mathbb{C}^{n\times n}$ which can be written as
$$ A = XD_AY^T \\
B = XD_BY^T $$
where $X$ and $Y\in\mathbb{C}^{n\times n}$ are unitary and with diagonal $D_A$ and $D_B\in\...
- 61
1
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2
answers
121
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How to solve the optimization problem $\max_{\mathbf{w}}\sum_i\text{sign}(\mathbf{w}^T \mathbf{x}_i)$?
I am looking for an algorithm to solve the following optimization problem
$$\max_{\mathbf{w}}\sum_i\text{sign}(\mathbf{w}^T \mathbf{x}_i)$$
where $\mathbf{w}$ and each $\mathbf{x}_i\in\mathbb{R}^d$.
...
- 175
0
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0
answers
123
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A quantity associated with an algebraic variete
Let $P:\mathbb{C}^{n}\to \mathbb{C}$ be an irreducible homogenous polynomial.
Is there a geometric or algebra geometric interpretation for the following quantity:
The maximum number $k$ such that ...
- 356
0
votes
1
answer
450
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Conjugacy in the quaternion group
Let $G$ be a non-commutative group, and suppose we are given two elements $x, y \in G$ which are conjugate, i.e. we know there exists some $z \in G$ such that $zxz^{-1} = y$. Can we find $z$ given $x$ ...
- 1,703
0
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0
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244
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Reduction from permanent to $(0,1)$-permanent and implication of $P \ne NP$
Valiant
shows reduction from counting the solutions of CNF formula $F$,$\#SAT(F)$
to computing permanent where $ Perm(A)= 4^{t(F)}\cdot \#SAT(F)$
for certain efficiently computable $t(F)$ and matrix $...
- 25.4k
0
votes
1
answer
454
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A determinant problem with symmetric PSD matrices
Suppose we have a a set of matrices in the complex field of the form $a_iv_iv_i^H$ for $i=\{1,\dots,n\}$ where $a_i$ are constant positive real scalars and $v_i$ are constant complex valued finite ...
- 213
0
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0
answers
173
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A combinatorial 0-1 matrix problem
Let $M \in \{0, 1\}^{n\times n}$.
Given a constant integer $c \ge 2$, let the number of $1$s in each row be equal to $n/c$ (assuming $c$ is a divisor of $n$).
Given a constant $\beta \in (0,1)$, we ...
- 1,677
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
0
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1
answer
154
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Dimension of a similarity class
Let $K$ be an algebraically closed field with characteristic $0$. I consider the Jordan decomposition of a NILPOTENT matrix: $A=diag(J_{r_1},\cdots,J_{r_s})\in M_n(K)$ where $J_k$ is the nilpotent ...
- 3,741
-2
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1
answer
1k
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Derivative of log determinant [closed]
Let $x_i \in\mathbb{R}^d$ and $a_i\in [0,1]$ for $i = 1,\dots,k$. How to compute the following derivative?
$$
\frac{d}{da_j}\log \det\left(\sum_{i = 1}^k a_ix_ix_i^\top\right).
$$
- 245