All Questions
2,027 questions with no upvoted or accepted answers
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What does this notation mean: matrix norm with a two-number subscript
I recently came across this notation, without explanation, in a paper:
$||\mathbf{W}||_{2,1}$
From the context, I know that $\mathbf{W}$ is a matrix, which could be any size, and that $||\mathbf{W}||...
- 111
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131
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Rings with the property $\dim R-\dim R/p\leq \text{const}$ for all minimal $p$
I'm curious if there exists a class of rings generalizing quasi-unmixed rings. I guess a generalization of quasi-unmixed rings can be done as follows:
For a fixed integer $i$
$$\forall p\in\...
- 189
1
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0
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158
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group of automorphisms of the Lie algebra of vector fields on affine variety
Hello,
Let $X$ be an affine variety and $A(X)$ be a ring of regular functions on $X$. Consider
a Lie algebra of derivations of $A(X)$ which we denote as $Der(A(X))$. It is known that $Aut(X)$ (group ...
- 143
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0
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122
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Standard equivalences and non-vanishing maps
EDIT : I edited the question according to Prof. Rickard's suggestions
Let $Y$ be an affine variety over $\mathbb{C}$ and $A$ and $B$ be $2$ algebras with finite homological dimension over $Y$ such ...
- 7,300
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0
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121
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General Characteristic Polynomial for Delay Matrices
Given a square matrix $A$ and a delay matrix $D(\lambda) = \textrm{diag}(\lambda^{m_1},\dots,\lambda^{m_N})$, where all $m_i$'s are integers.
I'm interested in the generalised characteristic ...
1
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0
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102
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Notion of transversality over the field of Puiseux series.
To a given a Laurent polynomial $f$ over the field of Puiseux seris with parameter $t$, $f \in \mathbb{C} \lbrace\lbrace t \rbrace\rbrace[z_1^{\pm1},...,z_n^{\pm 1}]$, one can associate the ...
- 41
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127
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clarify a question in group cohomology
In page 43 of Kenneth S.Brown's book "Cohomology of Groups", GTM 87, we have a proposition:
If $G=F(S)/R$ then there is an exact sequence $0\to R_{ab}\overset{\theta}{\to} \mathbb{Z}G^{(S)}\overset{\...
- 1,528
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0
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132
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Matrices with a common Fischer basis
Let $A$ be a real symmetric $n\times n$ matrix, normalized such that $Tr[A]=1$. Define a 'Fischer basis' as the basis in which all diagonal elements are equal to $\frac{1}{n}$. The motivation for ...
- 31
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97
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Degree of a commutator in a hyperalgebra or enveloping algebra
Consider a semisimple algebraic group $G$ over an algebraically closed field of arbitrary characteristic and let $\bar U(G)$ denote its hyperalgebra (ie, the restricted Hopf dual of the coordinate ...
- 3,637
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0
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781
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How to find the tensor product of modules that we don't know a basis for them?
Hi
I know how to calculate some easy tensor products like $\mathbb{Z}/m\mathbb{Z} \otimes_{\mathbb{Z}} \mathbb{Z}/n\mathbb{Z}\cong_{\mathbb{Z}} \mathbb{Z}/(m,n)\mathbb{Z} $ or $F[X] \otimes_{F} F[Y] \...
- 11
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306
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Annihilator ideals
For an ideal $I$ of a ring $R$ with identity, let $r(I)=\{r\in R: Ir=0\}$ and $l(I)=\{r\in R: rI=0\}$.
Question: If for any two ideals (two-sided ideal) $I, J$ of $R$, we have $l(I)+l(J)=l(I\cap J)$, ...
- 192
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0
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107
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Complementation in an extension field
If $E$ is an extension field of $F$, is $F$ necessarily (without assuming the axiom of choice) complemented as a vector subspace of $E$? (Of course the answer is easily yes if the extension is finite....
- 11.4k
1
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168
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Ring-theoretic version of a matrix problem
Problem #17 in Zhan's survey of open problems in matrix theory is the Li-Poon problem on writing a square real matrix as the linear combination of $k$ orthogonal matrices. They proved that it is ...
- 6,962
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237
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bivariate polynomial
Hello,
Let $p(x,y) = \sum_{m=1}^M\sum_{n=1}^N a_{m,n}x^{m-1}y^{n-1}$ be a bivariate polynomial where $\{a_{m,n}\}$ are complex.
If $(x_k, y_k), k=1,2,\cdots, MN-1$ are roots of $p(x,y)=0$ where $|...
- 9
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122
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the quasi-isomorphisms are homotopic-equivalence for DGAS
I just want to know more explainations on this result. So, please let me know where I can find the original paper on this result. THANKS
- 77
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0
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296
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Finding lower triangular matrix of an indefinite matrix
So I have the system $M = RS = RQQ^{-1}S $ and I have $R$ and $S$ currently.
I impose some constraints on $R$ in the form of $r^T$$QQ^Tr = 1$ where $r$ and $r^T$ are rows of R and their transposes. ...
- 11
1
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0
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221
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Centralizer in a matrix algebra over commutative polynomials
Let $A=M_n(F[x_{ij}\mid 1\leq i,j\leq n])$ be the matrix algebra over commutative
polynomials in $n^2$ variables $x_{11},\dots,x_{nn}$ over a (nice enough) field $F$.
I would like to know what is the ...
- 179
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115
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How many extreme maximal cliques are in an n*m 0-1 matrix?
We can use an n*m 0-1 matrix to denote a bipartite graph. Mining maximal bicliques in such matrix is an open problem.
The extreme maximal clique is a special maximal clique. A clique in such matrix ...
- 63
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144
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Structure groups and a special class of L-functions
Hello,
Let $X$ and $Y$ be two mathematical objects such that there exists a canonical embedding $f:X\hookrightarrow Y$. I define the structure group of $Y$ relatively to $X$, denoted $Str(Y/X)$, as ...
- 6,976
1
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263
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Average weighted value of a linear functional over increasing bounded subsets of Z^n
Say you're working within the finite-dimensional free Z-module $\mathbb{Z}^n$, and you want to impose a "norm" on this module. By a "norm" I mean a function $\|·\|: \mathbb{Z}^n \to \mathbb{R}$ which ...
- 4,897
1
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243
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Norm bound of the entrywise logarithm of a stochastic matrix stationary matrix
Hello,
Denote $\log_\star$ as the entrywise logarithm operation, and let $A$ be some row-stochastic matrix such that $\lim_{p\rightarrow\infty}A^p$ exists and all its entries are non-zero.
As a part ...
- 225
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165
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Coinduction and corestriction are quasi-inverse equivalences for comodules?
I'm reading http://arxiv.org/abs/math/0310337.
There the following statement is given without proof:
Let $k$ be a field. Let $C$ be a counitary coaugmented coalgebra, i.e. there is $\eta: C\to k$ ...
- 2,450
1
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0
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242
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separability of commutative rings
Before discussing on the main Question I should recall two notions in the area of commutative rings.
By $Max(R)$, we mean the set of all maximal ideals of the commutative unitary ring $R$.
...
- 1,788
1
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0
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72
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sharper interlacing
The usual interlacing inequalities say that if $M$ is a Hermitian $n \times n$ matrix and $\hat{M}$ is a principal submatrix of order $n-1$, then $\lambda_{\min}(M) \leq \lambda_{\min}(\hat{M})$. I ...
- 6,962
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86
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Finite dimensional consistently graded Lie superalgebras of depth greater than 2
Victor Kac, in the paper
"Classification of infinite-dimensional simple linearly compact Lie superalgebras", http://www.mat.univie.ac.at/~esiprpr/esi605.pdf
writes at the beginning of section 5 (p....
- 971
1
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0
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458
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semisimple algebra question
I know this is not a research level question, but may be I can get an idea...
Is there a direct proof of the following without going through composition series or Artin-Wedderburn theorem?
Let $V$ ...
- 103
1
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0
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158
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Comparing the volume of a rational lagrangian under a linear symplectomorphism.
Let's fix the standard symplectic structure $(\mathbb{R}^{2g}, \omega, J)$. A (marked) symplectic lattice then has the form $A\mathbb{Z}^{2g}$ for $A \in Sp_{2g}\mathbb{R}$. We say a vector subspace $...
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0
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628
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Totally unimodular Matrices
A matrix is totally uni-modular if the determinant of any (square) sub-matrix is {+1, 0, -1}. My question is, "Is there a way to transform(linear or non) a general matrix into a totally uni-modular ...
- 11
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0
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342
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Passing from Regular sequence to Prime ideal, for power sum symmetric polynomial
Let $S=\mathbb{C}[x_1,x_2,x_3,x_4]$ be a polynomial ring. Let $p_i=x_1^i+\cdots+x_4^i$ be the power sum symmetric polynomial in $\mathbb{C}[x_1,x_2,x_3,x_4]$.
Let $I=(p_1,p_2)$ be an Ideal of $\mathbb{...
- 446
1
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0
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199
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Noncommutative Localization "from scratch"
In his excellent monograph "Lectures on Modules and Rings" (GTM 189), T.Y. Lam remarks (10.13), p. 302, that there is no direct method of computing the kernel of the (co-)unit of the adjunction when ...
1
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0
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257
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level of rings and stable range of rings
The level $s(A)$ of a ring $A$ with unity $1$ is the smallest natural number $s$
such that $-1$ is a sum of $s$ squares in $A$. (If $-1$ is not a sum of squares in $A$, we
say that $s(A) =\infty$.). ...
- 11
1
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0
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190
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The smallest real part of eigenvales of weighted sum of two matrices
I have a matrix problem that I need help with.
Let H=aA+bB, where a+b=1,a>0,b>0,and A,B are matrices having non-negative real part eigenvalues. In addition, A+B has positive real part eigenvalues. I ...
- 135
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0
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2k
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Eigenvalues of Matrix Sum
Hello,
I have a linear algebra problem that I need help with.
Basically, I need to get the eigenvalues and eigenvectors of several (sometimes tens of thousands) very large Hermitian matrices (6^n x ...
- 11
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0
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182
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matrix-theoretic terminology query
Is there an accepted term for the following property?
Let $A$ be a real matrix such that all entries of the eigenvector corresponding to the least eigenvalue have the same sign.
NOTES: (1) The case ...
- 6,962
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179
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Splitting of noncommutative finite-dimensional local algebras over the residue field
Let $R$ be a finite-dimensional local (associative, unital, and not necessarily commutative) algebra over a field $k$ (that is, $R$ has a unique maximal two-sided ideal $\mathfrak M$) such that $\...
1
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0
answers
110
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The full linear ring is of finite projective dimension over the enevelopping algebra?
It is known that if $R=End_k(V)$, with $V$ a finite dimension $k$-vector space then $R$ is projective as $R^e$-module, thus of projective dimension $pd_{R^e}(R)=0$. If $V$ is of infinite dimension ...
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427
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Ring such that any submodule of an injective module is flat?
Does anyone know examples of rings $R$ with the property that any submodule of an injective (right) $R$-module is flat? If I'm not missing something, this class of rings includes the (Von Neumann) ...
- 21
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176
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On so-called self-covering matrices
(In this discussion I'm assuming all matrices are binary (0/1-valued).) We say that a matrix $M$ can be covered by another matrix $N$ if every entry in $M$ is either (1) NOT contained in $N$, or (2) ...
- 51
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140
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Diagonalizing matrices of linear forms of indeterminates
Let $B$ be a matrix with elements as linear forms of indeterminates. Is there a proper diagonalization procedure for such matrices like those of matrices with real and complex entries?
- 13.9k
1
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148
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Bounding the Schur's complement of similiar matrices
Assume the following:
• $L\leq K$
.
• $\Gamma\in M_{K,L}$ is a $L$ rank ${ 0,1} $ matrix, without identical rows or the zeros row.
• $N\in M_{K,K}$ is a diagonal matrix, whose diagonal is a ...
- 295
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0
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305
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tensor/hypermatrix analogues of $GL(n,\mathbb{C})$?
Please excuse me if this question turns out to be incredibly silly for one reason or another.
Are there tensor/hypermatrix analogues of $GL(n,\mathbb{C})$ that are interesting? What I'm mainly ...
- 1,075
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0
answers
155
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Koethe conjecture on non associative rings
Following up on Koethe conjecture I was wondering if the Koethe conjecture holds for non associative rings. My impression is that, it does not hold.But I am not sure ! Am I ...
- 11
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75
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Eigenvalues of matrices over a differential field
Let ($k$,$D$) be a differential field. Consider a square matrix $A$ with entries in $k$. We say that $A$ is $D$-similar to $A'$ if and only if there exists an invertible matrix $S$ such that $A$ = $S^{...
- 11
1
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227
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Joint Convexity of Spectral functions of several matrices
$\{A_1 \ldots A_K \}$ is a set of matrices in $\mathbb{R}^{m \times n}$. Let $f (A_1,\ldots,A_K)$ be a function of the singular values of all matrices. For e.g., $f$ is just summation of singular ...
- 519
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0
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417
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Absolute Irreducibility in Characteristic 2
Let $\mathbb F$ be a field and $\mathbb F[x_1,\dotsc,x_n]$ the ring of multivariate polynomials in $n$ variables over $\mathbb F$. A polynomial $P\in\mathbb F[x_1,\dotsc,x_n]$ is said absolutely ...
- 456
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0
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138
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Bases of Ideals With no Monomials
Let $K$ be an algebraically closed field and $K[\underline{x}]$ its ring of polynomials in $n$ variables $x_1,\cdots, x_n$. Let $J\leq K[\underline{x}]$ be an ideal such that there are no monomials in ...
- 345
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0
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688
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The sum of a nilpotent left ideal and a nil left ideal
In class, we recently saw that the sum of 2 two-sided nil ideals is a nil ideal. We were asked to show that the sum of a niplotent left ideal and a nil left ideal is a nil left ideal.
I am having ...
- 45
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0
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333
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Signature of quadratic form associated to an integral circulant matrix with only real eigenvalues
I am stil stuck with the following:
Let $C$ be a symmetric circulant matrix with integer coefficients of order $n=4k$
(e.g., $C=circ(-1,1,1,1)).$
Assume that $C^{-1}$ is a polynomial (with ...
- 1,641
1
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0
answers
500
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Dieudonné and generators of the orthogonal group
Let $k>0$ be a positive integer and $n=4k.$ A special case of a result of Dieudonné is
that every element $g$ of the orthogonal group in $n$ variables over the rational numbers
$$
G=O(n,\mathbb{Q}...
- 1,641
1
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0
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396
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Notation for bilinear form $y^t M z$, where $M$ is a matrix and $y,z$ are vectors.
I'm working on a problem where I need to consider a bilinear form of the form $y^t M z$ where $M$ is an $n$-by-$n$ real symmetric matrix and $y,z \in \mathbb{R}^n$ are vectors. I also need to consider ...
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