All Questions
1,923 questions with no upvoted or accepted answers
1
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0
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191
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Eigenvectors of contraction times projection
Suppose $A$ is a real $n\times n$ matrix with real eigenvalues:
$$
1=\lambda_1>|\lambda_2|\ge \ldots\ge |\lambda_n|>0.
$$
Suppose $B$ is an involution, for simplicity let us assume that
$B$ is ...
- 650
1
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0
answers
107
views
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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0
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192
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Non-negative Quadratic forms with Exterior Forms
Hello All,
I apologize if the following question is too elementary. Any suggestion is greatly appreciated. Thank you.
Let $n\geqslant 4$, $X$ be an $n$-dimensional inner product space over $\mathbb{...
1
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0
answers
290
views
Injective Mapping
Let y=Ax. A is a matrix n by m and m>n. Also, x gets its values from a finite alphabet. Elements of A and x can be complex numbers. How can i show if the mapping from x to y is injective for given A ...
- 11
1
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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$$QQ^Tr = 1$ where $r$ and $r^T$ are rows of R and their transposes. ...
- 11
1
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0
answers
220
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Dimension of faithful irreducible representations of $\mathbb{Z}_q\rtimes \mathbb{Z}_{p^2}$ in characteristic p,q
Let $p,q$ be primes s.t. $q=np+1$. Denote $m=p^2$. Then $\mathbb{Z}_p$ acts non-trivially on $\mathbb{Z}_q$, so we have a non-abelian semi-direct product $\mathbb{Z}_q\rtimes \mathbb{Z}_m$, with the ...
- 407
1
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0
answers
358
views
Kernel of modified Kronecker sum
The Kronecker sum of two matrices $A \in M(n \times n, \mathbb{R})$ and $B \in M(m \times m,\mathbb{R})$ is defined by the matrix
$$A \oplus B = A \otimes I_m + I_n \otimes B \in M(nm \times nm, \...
- 397
1
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0
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126
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Matrix Minimax problem
I have the equation $\Sigma_k(M_k{p_k})V=EV$, where the $M_k$ are n*n real Hermitian matrices, $V$ is a n*n eigenvector matrix, $E$ a dim-n energy eigenvector and the $p_k$ scalar parameters. The $M_k$...
- 4,829
1
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0
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880
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Order of column vectors in jordan normal form
When constructing the jordan normal form
of the matrix $A$ one has to construct the jordan blocks corresponding to the eigenvalues $\lambda_i$ of the matrix.
If the eigenvalue is simple or semi-simple ...
- 21
1
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0
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263
views
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 ...
- 4,907
1
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0
answers
243
views
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
1
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0
answers
475
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How does the Schmidt decomposition generalize to tensor products of several finite-dimensional systems?
Let $n,d_1,\ldots,d_n > 1$ be integers, and $V_1, \ldots, V_n$ be inner product spaces over $\mathbb C$, having dimensions $d_1, \ldots, d_n$ respectively. We consider the ways in which we may ...
- 1,444
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0
answers
192
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"Higher" Tangent spaces in char-p geometry - definition?
Hi, everyone!
I have some construction that requires exact definition.
I considered polynomial homomorphisms from $\Bbbk$ to $U_n(\Bbbk)$ (unitriangular matrices), where $char \Bbbk = p>0$ (more ...
- 1,422
1
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0
answers
109
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Is there a Krylov subspace method for solving D+epsilon*S where D is diagonal, epsilon small and S skew-symmetric
I'm working on a problem that gives a matrix system of the form D + epsilon*S, where S is a skew-symmetric matrix. I'm interested in finding if any work has been done to develop a conjugate gradient ...
- 11
1
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0
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109
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can minimal volume rational subspaces in a lattice be arbitrarily 'close'.
Let $\Gamma$ be a cocompact lattice in $\mathbb{R}^n$, eg. $\Gamma = A \mathbb{Z}^n$ for some $A \in SL_n \mathbb{R}$. Then any $k$-dimensional subspace $P$ which is rational in $\Gamma$ has a volume: ...
- 2,274
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 $...
1
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0
answers
413
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Combinatorial Interpretation of an Extension of Gaussian Polynomials
It is well-known that the Gaussian polynomial (or Gaussian coefficient, q-binomial coefficient) $\binom{n}{k}_q$ counts the number of $k$-dimensional subspaces of an $n$-dimensional vector space over $...
- 443
1
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0
answers
218
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Singular quadratic space
Let $(V,b)$ a symmetric bilinear space. An old theorem of Witt says that if $(V,b)$ is regular, then given a subspace $W$ of $V$ and an isometry $\sigma: W \to V$, there exists an isometry $\Sigma: V \...
- 5,163
1
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0
answers
629
views
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
1
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0
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443
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Diagonalizing matrix with a special conjugate transpose property
Hi all,
I'm looking for the minimum criterion on $A\in M_{3x3}(\mathbb{C})$ (a $3x3$ complex matrix) such that:
1) $A$ is diagonalizable by a matrix $T\in M_{3x3}(\mathbb{C})$
2) $T$ is such that $...
- 111
1
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0
answers
238
views
Classification of fibres in pencils of curves of genus two
For the case of characteristic positive, there exist some clasification of families of curves of genus two over a elliptic curve?.
- 11
1
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0
answers
212
views
componentwise eigenvector perturbation
Does the sin-theta theorem imply a componentwise nonasymptotic bound for eigenvectors? Assume, for the purpose of this question, that the eigenvalues concerned are simple.
If this is trivial, I ...
- 6,962
1
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0
answers
182
views
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
1
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0
answers
229
views
Counting equivalence classes in the transitive closure of two equivalence relations
Let $X$ be a finite set, and let $P_i$ and $Q_j$ be two partitions of $X$:
$$\bigsqcup_i P_i = \bigsqcup_j Q_j = X.$$
The finest partition which is nevertheless coarser than both $P$ and $Q$ is ...
- 5,898
1
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0
answers
285
views
Given a jointly convex function $f$, what is the bound of $f\left(\sum_ip_i^2x_i,\sum_jq_j^2y_j\right)$if $\mathbf{p},\mathbf{q}$ are constrained in a manifold?
Suppose there is a jointly convex function $f:\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}$, $\mathbf{x},\mathbf{y}\in\mathbb{R}^m$ and $\mathbf{p}=[p_1\;\ldots\;p_m]^\top,\mathbf{q}=[q_1\;\ldots\;...
- 607
1
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0
answers
216
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Characterizing symplectic matrices relative to a partial Iwasawa decomposition
Fixing notation: for matrices $A,X$ we let $A[X]$ denote ${}^tXAX$.
Let $P_n$ denote the collection of real $n\times n$ positive definite symmetric matrices.
For $Y\in P_n$ we have the usual ...
- 2,274
1
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0
answers
165
views
Affine space partition of a general set
Olof Heden, in his work "A survey of the different types of vector space
partitions", discusses various results regarding the following qustion - given a vector space $V$ over a finite field, how can ...
- 119
1
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0
answers
140
views
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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0
answers
298
views
Norm preserving matrix fix
Hello,
I'll state the problem first and than I'll a little bit of motivation.
Lets be given regular matrix $M \in \mathbb{R}^{n\times n}$ and norm $||.||$ in $\mathbb{R}^{n}$. Define $$ U =\{ L\in \...
1
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0
answers
538
views
Representing vertices of a cube using linear combination of tensor product of smaller cubes
Let $n,N \in \mathbb{N}$ with $N \ge n^{2}$.
Let $F[i] = \square[i]$ refer to the cube which has vertices from $\{-1,0,1\}^{n^{i}}$ ($n^{i}$ tuple of alphabets from $\{-1,0,1\} = \square[0] = F[0]$)
...
1
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0
answers
197
views
Matrix Theory approach to general Linear Equations over skew fields
Is there a matrix way of writing system of linear equation over a skew field where the variables in the equations are both left multiplied and right multiplied by elements of the skew fields.
Is the ...
- 113
1
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0
answers
128
views
Spectral decomposition of R matrix -> Wenzl projectors?
Just curious: if you take a R matrix from knot theory and apply
a spectral decomposition (see. e.g. my following post
Matrix decomposition the other way)
you'll get projectors: $T_i*T_j=T_i*\delta_{ij}...
- 4,829
1
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0
answers
222
views
(Non-)Surjectivity of the Maslov index
Let $V$ be a symplectic space over a field $k$ (for simplicity, the characteristic of $k$ is not $2$). The Maslov index sends a collection of $n$ lagrangian subspaces of $V$ to a quadratic space over $...
- 3,605
1
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0
answers
254
views
Operator eigenvalues and eigenvalues of pointwise evaluation matrix
Let $D\subset\mathbb{R}$ be a bounded interval and $f: D\times D \rightarrow \mathbb{R}$ a real-valued analytic function of two variables such that $f\in L_2(D\times D)$. Suppose we have upper bounds ...
- 3,217
1
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0
answers
213
views
Eigenvalue distribution of positive-definite analytic function
Let $f:[0,1]^2\rightarrow \mathbb{R}$ be a real-valued symmetric, positive definite function. Let $\{(x_i,y_i)\}_{1\leq i\leq N}\subset[0,1]^2$ be a finite, distinct set of coordinates. The point-...
- 3,217
1
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0
answers
606
views
how to prove uniqueness of matrix polynomial and its eigendecomposition
Hello, all!
Let $\underset{l \times l}{A(x)}$ be square polynomial matrix over $GF(q)[x]$, where $q$ is a prime power. Let $x_i \in GF(q^m)$ ($x_i \not= 0$) be eigenvalue of $A(x)$: $det(A(x_i)) = 0$....
spk
1
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0
answers
201
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What is the MP pseudoinverse's role in statistical learning and Self-Organizing Maps?
During a discussion in our lab last month, a professor mentioned to me that the behavior of Self-Organizing Maps can be described in terms of repeated applications of the Moore-Penrose psuedoinverse, ...
- 113
1
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0
answers
1k
views
Maximum of Tr$(ABA')$
Let $B$ be a fixed symmetric $M\times M$ matrix over the reals.
Let $A$ be an arbitrary $N\times M$ matrix over the reals.
I want to consider the problem of finding the extremal value of $\...
- 211
1
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0
answers
77
views
r-locally linear functions of many variables
Suppose $m,n,r$ are positive integers. Suppose V is a $m$-dimensional vector space over a field F. Let $G(V,n,r)$ denote the space of $n$-tuples of elements of V with the property that the vector ...
- 7,320
1
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0
answers
174
views
Eigenvalues of a Parametrized Family of Linear Functions
Suppose that we have a family of linear functions $L(\alpha) : \mathbb{R}^n \rightarrow \mathbb{R}^n$, where $\alpha$ is a positive real number.
For each $\alpha$, it is given that $L(\alpha)$ is a ...
- 111
1
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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
1
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0
answers
2k
views
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)?
$$
(Where both sides of the ...
- 637
1
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0
answers
265
views
"Lift and project" procedure for matrices
Definition. Let us call $n\times n$ matrix with non-negative entries good if sum of every row and column is equal to $1/n$.
Suppose we have a good matrix $A$. Let us consider the following strange "...
- 1,791
1
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0
answers
396
views
Notation for bilinear form $y^t M z$, where $M$ is a matrix and $y,z$ are vectors.
I'm working on a problem where I need to consider a bilinear form of the form $y^t M z$ where $M$ is an $n$-by-$n$ real symmetric matrix and $y,z \in \mathbb{R}^n$ are vectors. I also need to consider ...
- 4,922
1
vote
0
answers
169
views
Sum of two free o-submodules in a vector space over a local field
Let $V$ be a countably infinite dimensional $K$-vector space over a local field $K$ (nontrivially discretely valued with finite residue field). Let $o$ be the ring of integers of $K$.
Given two free ...
- 107
1
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0
answers
221
views
Nonunique low-rank matrix completion from a few entries
Suppose we want to have a good approximation for the following NP-hard problem
$$\min_{\bf X} \operatorname{rank}({\bf X}) \text{ s.t. } \mathcal{A}({\bf X}) = {\bf b}, {\bf X} \succeq 0$$
where ${\bf ...
- 449
1
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0
answers
1k
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Covariance matrix formula interpretation - what am I missing?
I'm reading a paper that outlines the calculation of a covariance matrix like the following:
$C=\displaystyle\sum^{N_b}_{i=1}\vec{x}_i\vec{x}_i^T$
What is the order of this matrix? My interpretation ...
- 111
1
vote
0
answers
393
views
iterated characteristic polynomials
If I have $N$ $M\times M$ symmetric positive definite matrices $A_i$ and an $N\times N$ positive semi-definite symmetric matrix B, let the $N\times N$ matrix $C_{ij}(\lambda)=B_{ij}$ for $i\ne j$ and $...
- 11
1
vote
1
answer
294
views
Hessian matrix of vectorized matrix product
I need to find the Hessian Matrix of $f(X,Y) = C \operatorname{vec} (A X^{-1} Y)$ where $C$ and $A$ are constant matrices and $X$ and $Y$ are the variable matrices. This would be a vector function of ...
- 11
1
vote
1
answer
1k
views
Spectral radius sum of two matrices
Let $W = S + cT$, where $|c| \le 1$ is a real constant and where $S$ and $T$ are square matrices containing real numbers from the interval $[0,1]$.
Assume moreover that the all eigenvalues $\lambda$ ...
- 11