All Questions
5,883 questions
0
votes
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443
views
Relation between degree of root of determinant polynomial and rank of the matrix
Let $A=[a_{ij}]$ be an $n \times n$ matrix with $a_{ij}=f_{ij}(x_1,...,x_m)$ where $f_{ij}(x_1,...,x_m)$ is a polynomial in $m$ variables over a finite field $\mathbb{F}_q$.
Let $rank(A)=n$.
Now ...
- 187
0
votes
1
answer
155
views
Are the inverses of a set of quadratic polynomials linearly independent?
Is a collection of reciprocals of monic reducible quadratic polynomials, that is functions of the form
$$ \{ \left( (x-a_i)(x-b_i) \right)^{-1} \}_{i=1}^{k}, $$
linearly independent over a finite ...
- 11
0
votes
1
answer
134
views
Modification of a known optimization problem
In my research of linear algebra and optimization, I wish to modify the following well-known problem:
$ \min \lVert x-Ax \rVert$ subject to $ rank(A)\leq k $ where $ x $ is a given column vector ...
- 620
0
votes
1
answer
111
views
A conjugation matrix $X\in \mathbb{C}^{ n\times p}$ where $p< n$
Given a Hermitian positive definite matrix $A\in \mathbb{C}^{n \times n}$ and a Hermitian matrix $B\in\mathbb{C}^{ p\times p},$ find the matrix $X$ so that $X^HAX=B$ holds where $X^H$ denotes ...
- 21
0
votes
1
answer
150
views
General results regarding linear separability?
I'm reading up on the theory behind support vector machines and would like a good reference with some general results about linear separability.
Specifically, questions like below:
Given two ...
- 101
0
votes
1
answer
75
views
Equivalent linear inequalities system - Coefficients bound?
Just having some difficulties with this system of inequalities...
We know E is a system of m linear inequalities of the form:
a1,1x1+ ··· +a1,nxn ≤ b1
...
am,1x1+ ··· +am,nxn ≤ bm
And E' an ...
0
votes
1
answer
110
views
Number theory for operator bound
Let $\gamma_i$ be such that for even $i$ $\gamma_i=1$ and for odd $i$ $\gamma_i$ shall have absolute value $1$ and the product of all of the odd ones is also on the complex unit circle but not 1 or -1....
- 501
0
votes
1
answer
327
views
Rotation matrix between two column spaces
I would like to find a rotation matrix between two flats $F_1,F_2$ that are defined by the column spaces of the matrices $M_1,M_2 \in \mathbb{R}^{n \times k}$ ($k<n$) respectively. If it was to ...
- 3
0
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1
answer
28
views
Finding a point at which only certain linear functionals are integral
Let $C$ be a full-dimensional rational polyhedral cone in $\Bbb R^d$ with facets $G_1,\ldots,G_n$ . For each $i$, let $h_i$ be an integer-valued linear functional on $\Bbb R^d$ whose kernel is the ...
- 3,079
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1
answer
556
views
Bounds on spectral norm of difference of powers of linear operators
Given bounded linear operators $A$ and $B$, with $||A-B||<\epsilon$, what is the tightest upper bound on $||A^n-B^n||$, for positive integer $n$? Additionally, for my specific situation, $A$ is ...
0
votes
1
answer
63
views
Nonnegative locus of the class of positive semidefinite symmetric bilinear forms
Let $\mathcal{P}$ denote the class of positive semidefinite symmetric bilinear forms $A : \mathbb{R}^{n} \times \mathbb{R}^n \to \mathbb{R}$. Let
$$
X(\mathcal{P}) := \{(x,y) \in \mathbb{R}^n \times \...
user94803
0
votes
1
answer
835
views
Do these matrices have the same null space?
Let $[\theta_1,\theta_2, \dots, \theta_N]^\mathrm{T} \, \in \mathbb{R}^N$. The angles are not all identical (on the circle), i.e. $[\theta_1,\theta_2, \dots, \theta_N] \not \equiv c [1,1,\dots, 1]^\...
- 33
0
votes
1
answer
228
views
Matrix series as block matrix times vector of matrix multiplication
Let
$V_i \subset \mathbb{R}^n$ and $V_0 \supset V_1 \supset ... \supset V_i \supset ...$,
$A_i, B_i: V_i \rightarrow V_i$ be square non-symmetric positive definite matrices,
$Q_i:V_{i-1}\...
- 323
0
votes
1
answer
568
views
Perturbation theory for matrices
I encountered the following problem. Since this is somewhat not related to what I normally do, I wanted to know what the best estimates in this field are.
Let $A \in \mathbb{R}^{n \times n}$ be a ...
- 3
0
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1
answer
198
views
The eigenfunctions of an operator commuting with all rotations.
When reading the paper
E. Carlen, J. Geronimo & M. Loss: SIAM J. MATH. ANAL., vol. 40, no. 1, 327-374
I found an argument like the following.
Given an bounded and self-adjoint linear operator ...
- 199
0
votes
1
answer
2k
views
How to calculate the inverse of the sum of kronecker products with the identity matrix
How to calculate $G^{-1}$ efficiently when $G$ is a large matrix knowing that:
\begin{eqnarray}
G=I⊗A + A⊗I
\end{eqnarray}
Or since i'm using $G^{-1}$ to multiply by some other matrix, how to find $...
- 21
0
votes
1
answer
142
views
sub-space restricted minimum eigenvalue of Hadamard product of two PSD matrices
Let $\mathbf{A},\mathbf{B}\in\mathbb{R}^{n\times n}$ be two positive semidefinite matrices. Also let $\mathbf{A}\circ \mathbf{B}$ denote the Hadamard product of $\mathbf{A}$ and $\mathbf{B}$. A ...
- 859
0
votes
1
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572
views
Recurrence Equation and Matrix Convergence
To begin with, let us give the conceptual background needed to expose the problem. First of all, we shall consider the set $\mathbb{L}^{n} = \mathbb{R}^{n}_{\geq0} = \{\overrightarrow{x}\in\mathbb{R}^{...
user89024
0
votes
1
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170
views
Non-strict column diagonally dominant matrix inner product
Let $A \in \mathbb{R}^{n \times n}$ be a normalized non-strict column diagonally dominant matrix, that is:
$$a_{j,j} = \sum_{i \ne j} \left|a_{i,j}\right|$$
where $0 \le a_{j,j} \le 1$ and $-1 \le ...
- 323
0
votes
1
answer
2k
views
SVD alternatives for symmetric matrices
Given any symmetric real valued matrix $A \in \mathbb{R}^{n\times n}$, I can decompose $A$ as the product of two complex matrices
$$
A = E'E
$$
Practically this can be done easily using SVD ...
0
votes
1
answer
174
views
The effect of linear transformation on generic vectors [closed]
I have a question about the effect of applying a linear transformation $M$ in $\mathbb{R}^{n \times n}$ to a vector $v \in \mathbb{R^n}$.
I know that if $M$ has p-norm $\|M\|_p = \lambda$, then by ...
- 371
0
votes
1
answer
535
views
a sum of ratios of quadratic forms
I have the following function that I would like to optimize over the value A
$$f(A)=\sum_k \frac{\mathbf{y}_k^H\left[\begin{array}{cc} 1&0\\ 0& A \end{array} \right]\mathbf{x}_k\mathbf{x}_k^H\...
- 61
0
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1
answer
286
views
Are $\left[\begin{matrix}x_\ell \\ x_\ell\varphi_k^\ell\end{matrix}\right]$ linearly independent?
Let $\varphi_k\in\mathbb{C}$ be a primitive $k$-th root of unity, and define the sets
$$S_\ell:=\left\{\left[\begin{matrix}x\\x\varphi_k^\ell\end{matrix}\right]\in\mathbb{C}^{2n}\;\middle|\;x\in\...
- 271
0
votes
2
answers
149
views
Simultaneous special orthogonal similarity problem
Given matrices $A,B,C,D\in\Bbb K^{n\times n}$ where $\Bbb K$ is a ring is there an efficient technique to compute set $O$ with $OO'=I$ where $'$ is transpose and $\mathsf{Det}(O)=\pm1$ such that
$$A=...
user76479
0
votes
2
answers
588
views
Estimating the shift in the $\lambda_{\max}$ of a matrix under a diagonal perturbation
Given a matrix $A$ and a diagonal matrix $D$, how can we estimate $\lambda_{\max}(A+D) - \lambda_{\max}(A)$? Feel free to make other assumptions about the matrices that they are all symmetric and have ...
0
votes
1
answer
371
views
Is there relation between vector valued RKHS and interpolation space?
Vector valued RKHS which is covered extensively in the book "Pick Interpolation and Hilbert function spaces" . In a different context interpolation space is defined in the wikipedia link: https://en....
- 495
0
votes
2
answers
1k
views
The condition number of a scaled Vandermonde matrix
Let $V(x_1,..,x_n)$ be the Vandermonde matrix induced by $x_1,..,x_n$, and
let $\tilde{V} := V(\frac{x_1}{h},...,\frac{x_n}{h})$.
My intuition says that the condition number should be invariant under ...
- 265
0
votes
1
answer
122
views
Solving $\lambda U^{\dagger}V -\bar{\lambda} V^{\dagger}U = A$ [closed]
Given an $A \in \mathfrak{su}(n)$, is it always possible to solve for $U,V \in SU(n)$ and $\lambda \in \mathbb{C}$ such that $\lambda U^{\dagger}V -\bar{\lambda} V^{\dagger}U = A$?
Cross posted from ...
- 2,099
0
votes
2
answers
114
views
Symmetric matrix from a nonsymmetricc matrix
Basically this is a part of a long algorithm to calculate some matrix properties.
Given an upper triangular square matrix R, how can I find an orthonormal matrix W (possibly iteratively) such that WR ...
0
votes
1
answer
124
views
Stationary distribution of random walk alias solving uncountably many linear equations [closed]
Let us have interval $I = (i_1,i_2)$, function $f_1 : I \mapsto I$, function $f_2 : I \mapsto I$.
Let $x_0$, $x_1$, $x_2$, ... be series of random variables from interval $I$ denoting random walk. ...
0
votes
1
answer
72
views
Characterisation of a matrix ordering property
Let $n$ be a positive integer; we consider all matrices mentioned henceforth to be $n$-by-$n$ matrices. Let $A$ and $B$ be matrices wherein all entries are nonnegative (such matrices will be called ...
- 345
0
votes
1
answer
118
views
Action of $(\mathbb{Z}/2g\mathbb{Z})$ on quadratic forms on $\mathbb{Z}/2\mathbb{Z}$-vector space
Let $\mathbb{Z}/2\mathbb{Z}$ the 2 elements field, with additive notation.
I need some clarifications on the relation between quadratic forms on a $\mathbb{Z}/2\mathbb{Z}$-vector space (say, of ...
- 3,779
0
votes
1
answer
456
views
Integral over Kronecker product
Let $A : [0,T] \to \mathbb R^{n \times n}, t \mapsto A(t)$ be smooth with the property that
$$ \int_{0}^T A(t) dt $$ is invertible.
Does it then follow that the matrix
$$ \int_{0}^T A(t) \otimes A(...
user45183
0
votes
1
answer
1k
views
How do eigenvalues change if we duplicate a row and column of a symmetric matrix?
Let $\bf A$ be a $n \times n$ symmetric positive semidefinite matrix whose first column is denoted by ${\bf a}_1$. We define a new matrix,
$$ {\bf B} = \begin{bmatrix} a_{11} & {\bf a}_1^T \\ {\bf ...
0
votes
1
answer
546
views
Solution of infinite dimension linear system
Suppose that ${a_n}$ and $b_n$ is decreasing sequence such that $a_0=A$, $lim_{n->\infty}a_n=0$ and $b_0=B$, $lim_{n->\infty}b_n=0$.
For fix n,
we can construct n dimension linear equation ...
- 111
0
votes
1
answer
664
views
Kneser graphs eigenvalues
Basically, I want to prove that, in the Kneser graph (wikipedia has a good definition),$K_{n, m}$, if $n_{-}(A(G)) $ and $n_{+}(A(G))$ denote the number of negative and positive eigenvalues of A(G) ...
0
votes
1
answer
203
views
Eigenvalues of a given parametrized matrix.
Let $\mathbf{A}$ and $\mathbf{B}$ be two complex rank-one $N\times N$ positive semi-definite matrices. Let the matrix $\mathbf{C}$ be defined as
\begin{align}
\mathbf{C}=\left(\mathbf{I}*\frac{1}{\...
- 1,421
0
votes
1
answer
720
views
Perturbation of Cholesky decomposition for matrix inversion
I am looking for a computationally cheap way to compute $x$ such that $$(L L^T + \mu^2 I)x = y$$
where $L \in \mathbb{R}^{n \times n}$ is a lower triangular definite positive matrix (with some very ...
- 103
0
votes
1
answer
151
views
Measuring the extent of entanglement in tensor products
Is there a non-negative integer valued grading function on the tensor product of two Hilbert spaces which measures the extent of entanglement ?
0
votes
1
answer
157
views
singular values function
Let $\mathbf{F}\in\mathbb{C}^{M\times M}$ and $\mathbf{D} = \operatorname{diag}(\mathbf{d})$ where $\mathbf{d}\in\mathbb{R}^M$. By SVD, $\mathbf{F}\mathbf{D}\mathbf{F}^H=\mathbf{U}\mathbf{S}\mathbf{U}^...
- 9
0
votes
1
answer
2k
views
eigen-decomposition solution? is it unique?
Assume an N*N covariance matrix (Q) which is a positive definite matrix. The decoder X is assumed to be N*s, where s<=N. X is calculated to be s eigenvectors corresponding to s minimum eigenvalues. ...
- 1
0
votes
1
answer
242
views
does the basis in the singular value decomposition of a sum depend on the singular values of the summands
Suppose you have 4 matrices with singular value decompositions
$A = U_1 \Sigma_A V_1^{\dagger}$, $B = U_2 \Sigma_B V_2^{\dagger}$, $C = U_1 \Sigma_C V_1^{\dagger}$ and $D = U_2 \Sigma_D V_2^{\dagger}$ ...
0
votes
1
answer
105
views
The influence of eigendecomposition on the periodicity of a (rank 2) Hermitian matrix (of functions)
Let $\boldsymbol{R}(u,v);~u,v\in\mathbb{R}$ be a Hermitian matrix (of Hermitian functions) with entries
\begin{equation}
r_{ij}(u,v) = 1 + Ae^{-2\pi i \phi_{ij}(ul_0 + vm_0)}; A\in\mathbb{R},l_0\in\...
- 33
0
votes
1
answer
222
views
How do you solve a tridiagonal matrix where all 3 diagonals are ones? [closed]
This is probably really simple, and I'm missing something, but Thomas' algorithm doesn't seem to work.
- 1
0
votes
1
answer
520
views
solving trace norm equality [closed]
Problem Formulation
under what conditions can we solve $\mathrm{trace}(\mathbf{AB})=0$ ? or more specifically, when will $\mathrm{trace}(\mathbf{AB})=0$ implies that $\mathrm{trace}(\mathbf{B})=0$.
...
- 125
0
votes
2
answers
174
views
Matrices whose kernel escapes a sub-vector space
Let $n>n'\gg m$ and $V$ be a subspace of $\mathbb{C}^n$ of dimension $n'$. I am trying to characterize the set $X$ of $m\times n$ matrices $A=(a_{ij})$ satisfying $\ker(A)\not\subseteq V$, that is, ...
- 537
0
votes
1
answer
170
views
Tensoring with descending chain of modules
Let $A \to B$ be a ring homomorphism. Let $M_1 \supseteq M_2\supseteq \ldots$ be an infinite chain of $A$-modules ($M_i$ not necessarily finite free). Suppose that the limit $\cap_{i=1}^{\infty} M_i$ ...
- 11
0
votes
1
answer
172
views
orthogonality in a lattice
Let $\Lambda$ be a lattice with a quadratic form $q$ of signature (3,19).
Let $\Lambda_{\mathbb{R}}:=\Lambda\otimes \mathbb{R}$ and $W\subset \Lambda_{\mathbb{R}}$ a positive subspace of dimention 3.
...
- 107
0
votes
1
answer
139
views
Spectrum of a Laplacianized matrix
Suppose that $A$ is a positive matrix and that we let $R$ be the diagonal matrix of $A$'s row-sums. What can be said about the spectrum of $R-A$? I am particularly interested in the largest eigenvalue ...
- 6,962
0
votes
2
answers
573
views
about decomposition of three forms
Patrick D. Baier in his Ph.D. thesis for proving the theorem 2.1.4 used the following non-trivial fact (in chapter 2 on page 14):
Let $0\neq X\in V$ (here $V$ is of dimension 6), $W^\ast = Ann(X)$ ...
user21574