Questions tagged [linear-algebra]
Questions about the properties of vector spaces and linear transformations, including linear systems in general.
5,874 questions
0
votes
1
answer
365
views
How to estimate the norm of a matrix
There is a matrix as following,
\begin{eqnarray}
A = \left (
\begin{array}{l}
0 \quad \quad \quad \quad \quad \quad \quad ~~ 1\\
b \quad ~~~0 \quad \quad \quad \quad \quad a\\
ab \quad ~~ b \...
0
votes
1
answer
711
views
a system of nonlinear equations (power sum)
greetings . is there a general method-algorithm to solve the following system !?
$\sum_{n=1}^{m} {x_{n}}^{j}= {k}_{j} $
$j=1,2,...,m$
$k_{j}$ are constants
thanks in advance
- 181
0
votes
1
answer
409
views
Need help to find an efficient algorithm for the following problem!
Consider $x$ an n-dimensional vector with $x_i$ is integer in the range $[0 \dots k], k\in N$.
Given $A_{n\times n}$ is the covariance matrix of $x$.
$u$ is a given n-dimensional vector of real ...
- 131
0
votes
1
answer
307
views
Comparing iterative methods for linear systems
For a tridiagonal matrix of the from
\begin{bmatrix}
a & -b & \newline
-b & a & -b \newline
& \ddots & \ddots & \ddots \newline
& & & & -...
0
votes
1
answer
1k
views
Conjugate Matrix
Let $A$ be a nilpotent square matrix, $J$ be the antidiagonal matrix with 1's on the secondary diagonal (i.e. $J^{2}=E$) and let $B=J A J.$ Suppose we conjugate the matrices $A,B$ by a matrix $...
- 301
0
votes
2
answers
415
views
Commutative *-subrings of the noncommutative C*-algebra $B(l^2)$
A $\star$-ring is a ring with an involutive anti-automorphism. The simplest example of a noncommutative $\star$-ring is perhaps $B(l^2)$, the ring of bounded linear functions on the sequence space $l^...
user2529
0
votes
1
answer
249
views
Going from individual elements back to to matrix/vector notation [closed]
Note: Moved to math.stackexchange.com. Sorry for the off-topic question!
[Context: I'm working through problem set one of Andrew Ng's Machine Learning course, question 2, trying to derive the Hessian ...
- 101
0
votes
1
answer
322
views
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 ...
- 449
0
votes
1
answer
2k
views
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 ...
- 225
0
votes
1
answer
263
views
Separability of inner product to a product of Minkowski function and norm
I’ve encountered the following assumption:
Let D be a set such that there exists a Minkowski function $f(u)$ on $\mathbb{R}^l$ and norm $g(v)$ on $\mathbb{R}^m$ such that
$\forall u\in \mathbb{R}^l, \...
- 1
0
votes
1
answer
406
views
Operation of GL_n(Z/bZ) [closed]
I want to show, that $GL_n(\mathbb{Z}/b\mathbb{Z})$ operates transitively on
$X = \{ (v_1, \ldots, v_n) \in (\mathbb{Z}/b\mathbb{Z})^n \ | \ v_1\mathbb{Z}/b\mathbb{Z} + \ldots + v_n\mathbb{Z}/b\...
- 3
0
votes
0
answers
46
views
max eigenvalue and schatten-1 norm of depolarizing channel acting on trace-0 Hermitian matrix
Denote $\mathcal{H}^n$ as the $n-$dimension Hermitian matrices. Depolarizing channel $\Delta_p:\mathcal{H}^2\to\mathcal{H}^2$ is defined as $\Delta_p(A)=p\text{ tr }(A)~I/2+(1-p)A$ where $A\in \...
- 91
0
votes
0
answers
118
views
Uncomplete argument in Nishioka book
In Nishioka book "Mahler functions and transcendence" in the proof of Theorem 4.2.1, Nishioka asserts the following: For a matrix $A=(a_{i,j})_{1\le i\le m}$ with coefficients in $K[z]$ ($K$ ...
- 3,998
0
votes
0
answers
87
views
Is there a name for "applying linear operations to vector sequences from the right"?
Let $v_1,...,v_n\in\Bbb R^d$ be a sequence of vectors. When we say that we "linearly transform" this sequence, we mean that we apply a linear transformation $T\in\Bbb R^{d\times d}$ to each ...
- 13.6k
0
votes
1
answer
48
views
How to efficiently replace the Paterson-Stockmeyer algorithm and reduce non-scalar multiplications
Paterson-Stockmeyer algorithm
If we need to compute a high-degree polynomial expression, such as:
$$
P(y) = \sum_{k=0}^{B} a_k y^k
$$
the Paterson-Stockmeyer algorithm can process the powers in ...
- 3
0
votes
0
answers
66
views
Taking trace of a tensor product of matrix-valued smooth functions on the thin diagonal
Let $V$ be a finite dimensional real / complex vector space and consider the space $L(V,V)$ of linear operators on $V$.
Fix $n \in \mathbb{N}$ and let $\mathcal{M}$ be the real / complex vector space ...
- 3,477
0
votes
0
answers
50
views
Eigenvalues of functions on finite discrete sets
Suppose I have an arbitrary function on a finite and discrete set $S$ defined as
$$f: S \times S \to \mathbb{C}^{|S|\times |S|}.$$
The $|S| \times |S|$ matrix $M$ is defined as
$$(M)_{ij}=f(s_i, s_j) \...
- 1
0
votes
0
answers
60
views
The generalized Laplace expansion for tensor
I'm reading this paper https://arxiv.org/abs/1308.3860.
In the Appendix (page 22), the author uses a generalized Laplace expansion for the determinant tensor, as shown in the picture1.
But I only ...
- 1
0
votes
0
answers
51
views
Degree of determinant of a (non-monic) matrix polynomial
Let $n=2, 3, \dots$ and consider the matrix polynomial $L(\lambda)=\sum_{k=0}^{\ell}A_k\lambda^k$, where $A_k \in \mathbb{C}^{n\times n}$.
In the so-called monic case (or that can be made monic by ...
- 11
0
votes
0
answers
28
views
Constructing random graphs with given eigenvalues and eigenvectors
In Linial's presentation on SOME PROBLEMS AND RESULTS IN THE
GEOMETRY OF GRAPHS, on slide 7, some relations of properties of graphs to the eigenvalues of their adjacency matrix are listed, e.g.
if $G$...
- 13.2k
0
votes
0
answers
15
views
Change in two spectral deviations due to edge deletion in a signed graph
Prove (or disprove) the following. Let $\Sigma=(G,\sigma)$ be a given signed graph. If $\lambda_1\ge\lambda_2\ge\cdots\ge \lambda_n$ and $\mu_1\ge\mu_2\ge\cdots \ge \mu_n$ are the eigenvalues of the ...
- 61
0
votes
0
answers
46
views
What's the problem in using spanning Bessel sequences that are not frames to decompose vectors?
This is related to a question I recently asked on math.SE.
Consider a subset $G\equiv \{g_k\}_{k\in\mathbb{N} }\subseteq\mathcal H$ in a separable Hilbert space $\mathcal H$, and suppose $G$ spans the ...
- 342
0
votes
1
answer
140
views
Finding positive vectors of a special LGS
Let the following $4 \times 4$ LGS be given for which all coefficients
$a_1, a_2, a_3, a_{11}, a_{12}, ..., a_{33}$ are $>0$:
$a_1 + a_{11} \; x_1 + a_{12} \; x_2 + a_{13} \; x_3 + 0 \; x_4 = (a_{...
0
votes
0
answers
121
views
Closed form of coefficients of a finite field polynomial
I want to find a valid polynomial for a finite field $\mathbb{Z}_p[x]_{f(x)}$ with $d=deg(f(x))$. For this definition to hold, it can be deduced that $p$ must be prime and the polynomial $f(x)$ ...
- 47
0
votes
0
answers
68
views
Inequality between product of companion matrices and power of Pisot number
Let $d\geqslant 2$ be an integer and consider a convergent sequence of "companion" matrices
$$A_k := \begin{pmatrix}
a_{k,1} & a_{k,2} & \cdots & a_{k,d} \\\
& ...
- 101
0
votes
0
answers
94
views
Infinite sequence of PSD non-moments in two variables
Define a 2d sequence to be a mapping $a: \mathbb{N}^2 \to \mathbb{R}$ (where $\mathbb{N} = \{0, 1, \dots\}$). Here are two definitions of types of 2d sequences:
We say that a 2d sequence $a$ is a ...
- 161
0
votes
0
answers
52
views
What are the injective embeddings of R^d into the cone of (semi-) positive definite matrices of dimension d?
How can we characterize the set of all injective functions from $\mathbb{R}^d$ to the set of all symmetric positive definite matrices of dimension d?
- 109
0
votes
0
answers
70
views
Cyclotomic eigenvalue question for Distance-regular graph
I have read this paper. So, I am just thinking about if the following guess is true:
GUESS: Any Distance-regular graph (DRG) has cyclotomic character value property (which means the eigenvalues of a ...
- 109
0
votes
1
answer
158
views
Techniques for bounding the operator norm of the expectation of random matrix?
Let $\mu$ be a distribution on the unit sphere in $\mathbb{R}^n$. Let $u \sim \mu$ and consider the random matrix
$$
A = I_n - uu^T.
$$
Question: What techniques are available to provide (reasonably ...
- 420
0
votes
0
answers
51
views
Minimizer of forward and reverse Kullback-Leibler divergence with sum constraints on marginals
Consider minimization of the Kullback Leibler divergence between two discrete distributions $p$ and $q$:
\begin{align*}
D_{KL} \left( p \parallel q \right) = \sum_i p_i \log \left( \frac{p_i}{q_i} \...
0
votes
0
answers
31
views
What is the Fisher information matrix of the von Mises-Fisher distribution?
Assuming the von Mises-Fisher distribution as
$$f_{p}(\mathbf{x}; \boldsymbol{\mu}, \kappa) = C_{p}(\kappa) \exp \left( {\kappa \boldsymbol{\mu}^\mathsf{T} \mathbf{x} } \right),$$
where $\kappa \ge 0$,...
- 287
0
votes
0
answers
96
views
When can a point be reconstructed from relative angle measurements?
Given a set of points $p_1,\dots,p_n$ in $\mathbb{R}^d$ and a target point $x\in\mathbb{R}^d$, I measure all the angles between all pairs of points and the target point. In other words, I have the ...
- 427
0
votes
0
answers
22
views
Eigenvalues of Composition of Hadamard Operations of Low Rank Matrices
I am interested in the eigenvalues of $$ee^T \oslash (aa^T - a^{\odot2}(a^{\odot2})^T )^{\odot \frac{1}{2}},$$ where $a \in \mathbb{R}^n$ and $e$ is the vector with all entries equal to one.
Can we ...
0
votes
0
answers
93
views
Orthogonalization of symmetric non-degenerate bilinear forms
It is well-known that given a field $k$ with characteristic different from $2$, every symmetric non-degenerate bilinear form $B$ over a finite-dimensional space can be orthogonalized. This means that ...
- 1,485
0
votes
0
answers
43
views
Absolute value of elements of b=Ax and the minimum singular value of A
For $b=Ax$, is there a way to relate the minimum absolute value of the element of $b$, $\min|b_i|$, and the minimum singular value, $\sigma_\text{min}$, of $A$?
What I want is something like: $\sigma_\...
0
votes
0
answers
40
views
Eigendecomposition of Toeplitz matrix
I am working with Toeplitz matrix, I know that a Toeplitz Matrix $T$ can be decompose as a sum of a circulant and skew circulant matrix which can be diagonalized using the DFT matrix
$T=C+S = F\...
- 1
0
votes
0
answers
129
views
Linearly independent Kronecker product construction
I have a question regarding a constructive argument about Kronecker products which came up while trying to solve a more general problem.
Let $n\in \mathbb{N}$ and $E \subseteq [n] \times [n]$ with $d^...
0
votes
0
answers
32
views
Finding measure representation for rank 2 moment matrices
Assuming the following equation has a solution, I'm interested in finding any concrete values of $x_{1},\dots x_{n},y_{1},\dots y_{n},c_{1},c_{2},R$ that fulfills it.
$$
\begin{bmatrix}
1 & 1 \\
...
- 275
0
votes
0
answers
101
views
Eigenvectors of tridiagonal hermitian matrix
In my paper, I investigate the coordinates of the eigenvectors of a hollow tridiagonal hermitian matrix, which is defined as:
\begin{align*}
Q_n =
\begin{pmatrix}
0 & q_{1,2} & 0 & 0 & ...
- 1
0
votes
0
answers
77
views
Eigenvalues of N×N correlation matrices as N tends to infinity
I want to find a 𝑁×𝑁 positive definite correlation matrix, which ensures that as 𝑁 goes to infinity, only a finite number of eigenvalues remain non-zero, while the rest eigenvalues approach zero.
...
0
votes
0
answers
109
views
Linear independence in $\mathbb{Z}_q^n$
Consider $\mathbb{Z}_q \equiv \mathbb{Z}/q\mathbb{Z}$, where $q \geqslant 2$. A set of vectors in $\mathbb{Z}_q^n$ is said to be linearly independent if no nontrivial linear combination of them ...
- 503
0
votes
0
answers
19
views
Equality Issue in Deriving Covariance Update for Kalman Filter
I am currently working on deriving the Kalman Gain from the covariance of the updated state and have encountered an equality issue that I am unable to resolve. Below are the derivation steps and the ...
0
votes
0
answers
72
views
Minimizing the Spectral Norm of the Hadamard Product of a Quadratic Form Using CVX
I am trying to use CVX to minimize the spectral norm of the Hadamard product of two matrices, one of which is in quadratic form. Specifically, I am trying to minimize $\|{\bf A} \odot {\bf XX}^H\|_2$, ...
- 43
0
votes
0
answers
38
views
Sequence of projections that alters a $2^d$ tuple of points to a hyperparallelepiped
Suppose we have a $2^d$ tuple $\{ x_i \}_{i=0}^{2^d-1}$ of points in some $\mathbb{R}^n$. I would like to shift the points of this tuple in some controlled way, so that the final $2^d$ tuple $\{ y_i \}...
- 820
0
votes
0
answers
123
views
Realizable singular value spectra of normalized finite frames
$\DeclareMathOperator\tr{tr}$Let $m, n \in \mathbb{N}$, $m \geq n$, and let $\{f_i\}$, $1 \leq i \leq m$, be $m$ unit vectors (wrt. 2-norm) in $\mathbb{R}^n$. Let $A = [f_1 \, \, \, f_2 \, \, \, \...
0
votes
0
answers
36
views
Conjugate gradient-like algorithm with multiple search directions
I am solving an $n*n$ system $Ax=b$ in CUDA where $A$ is a sparse matrix. Currently I am solving it using the conjugate gradient algorithm.
I have noticed that $Ax$ where $x$ is $n*1$ has roughly the ...
- 71
0
votes
0
answers
47
views
Counting zero-sum subsets of a finite field with a particular form
Let $\mathbb{F}$ be a finite prime field of characteristic different than $2$ and $\beta \in \mathbb{F}$ a generator of the $2$-power order multiplicative subgroup of order $2^k$, so $\beta^{2^{k-1}} =...
- 3,058
0
votes
0
answers
66
views
Random elliptical potential lemma
Elliptical Potential Lemma: Let $V_0 \in \mathbb{R}^{d \times d}$ be positive definite and $a_1,a_2,...,a_n \in \mathbb{R}^{d}$ be a sequence of vectors with $||a_t ||_2 \leq L < \infty$ for all $t ...
- 61
0
votes
0
answers
28
views
Example of a matrix -HDH that is not PSD (with non-euclidean distances D)
It's widely known that, given a matrix of squared Euclidean distances, $\mathbf{D}_{ij} = \| \mathbf{X}_i - \mathbf{X}_j \|^2$, and the centering matrix $\mathbf{H} = \mathbf{I} - \dfrac{1}{n}11^T$, ...
- 101
0
votes
0
answers
32
views
Eliminating nullity for enhanced non-singularity
If we have an
$n\times n$ matrix $A$ with entries either $0$ or $1$, where all diagonal entries are $0$ and the rank is $k<n$, can we reach full rank by changing exactly $n-k$ zero off-diagonal ...
- 4,058