Questions tagged [linear-algebra]
Questions about the properties of vector spaces and linear transformations, including linear systems in general.
5,874 questions
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Differential form of the multidimensional "orthogonal dilation" operator
For a one-dimensional $f(x)$, the dilation operator $f(ax)$ can be expressed as $\exp(g(D))f(x)$, where $g$ is a closed-form function. This is easily checked by e.g. formal Taylor series expansion.
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Approximation for an expectation expression
Let $\mathbf{x} \in \mathbb{C}^M$ is an unknown distributed random vector (certainly not gaussian), and matrix $\mathbf{A}\in \mathbb{C}^{M \times M}$ which is fix (known). Also, assume we know the ...
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Maximum number of vectors with bounds on inner products (follow up question)
This is a follow-up question from my previous question.
Suppose there are (2n+1) vectors $\{m_1,m_2,...,m_n\}$, $\{p_1,p_2,...,p_n\}$ and $p^*$ in $R^{k+1}$. $m_i$ are weakly positive vectors. $p_i$ ...
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142
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Vandermonde matrix with polynomials
Let us consider the simple Vandermonde matrix $V_n$ with $V_{ij} = \omega^{(i - 1)(j - 1)}$ where $\omega = e^{2i\pi/n}$. Its well known that for a column vector $A$, $VA$ is equivalent to evaluating ...
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75
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The spectrum of the product $JA$ where $J=I_n\oplus (-I_n)$
Let $A$ be a real symmetric matrix in $M_{2n}(\mathbb{R})$with $A^2=I_{2n}$. Suppose that the Schur decomposition of $A$ is given by $A=\Lambda^t D \Lambda$. Let us consider the following matrix.
$$...
- 4,058
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Is there a redundant constraint in linear programming? [closed]
From wikipedia:
But... Why do we need the $x\ge 0$ part? We can instead do $-x\le 0$, and thus saving a line in the definition (which is not a big deal but nevertheless nice).
(In order to do that, ...
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257
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Calculation of solid angle for rectangle in 6DOF [closed]
I am an undergrad trying to understand and use solid angle calculations:
I have a point source in R3 space (x_source, y_source, z_source) and a rectangle with given center (x_center, y_center, ...
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80
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Probability of accurate sparse recovery
Suppose $\mathbf{A}_{k\times n}$ ($k<n$) is a matrix whose entries are generated i.i.d. from Gaussian distribution and $\mathbf{s}_{n\times 1}$ is a sparse vector with $m$ sparsity (i.e., $\|\...
- 287
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Analytic formula for minimizer of $f(x) := \sqrt{(x-a)^\top S(x-a)}+ r \|x\|_2$
Let $S$ be a positive-definite $n \times n$ matrix and define $\|z\|_S := \sqrt{x^\top S x}$ for any $x \in \mathbb R^n$. Let $a$ be a fixed vector in $\mathbb R^n$ and $r \ge 0$, and consider the ...
- 6,853
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Product of subspace and its inverse
$\DeclareMathOperator\GF{GF}$Let $R=\GF(q)$ be a finite field with $q=p^r$ elements, where $p$ is a prime number, $S=\GF(q^n)$ be an extension of $R$, where $n\in \mathbb{N}$, $n\geq 2$ and let $K=\GF(...
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116
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Enumerating (i.e. generating one by one) matrices of given rank over a finite field
Let be given positive integers $m,n,r$, with $r \leq \min(m, n)$, and a finite field of $q$ elements $\mathbb{F}_q$.
I'm looking for an efficient algorithm to enumerate (i.e., generate one by one) all ...
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Explicit expression of Padé–Hermite approximant of type I
It is well known that the Padé approximants $(P,Q)$ of an analytic function in the neighborhood of $0$ can be expressed as a quotient of Hankel determinants built on the coefficients of the function $...
- 3,998
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Special type of normal form of matrix in principal ideal domain
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\PSL{PSL}$I want to ask the following, Given $X \in n \times n$ matrix that all the elements are integers and $X=X^{T}$ is symmetric.
Can one always ...
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Order 2 matrices with entries in the polynomial ring over a field are diagonalisable
This is a variant on the question posed here, in which the OP asks for a characterisation of the diagonalisable involutions in $\operatorname{GL}_n(A)$, where $A$ is a $k$-algebra for some field $k$ ...
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Maximum vertex amount of low-dimensional simplex projection
Consider an arbitrary simplex $\mathcal{S} \subseteq \mathbb{R}^n$ ($\mathcal{S}$ is a polytope in $\mathbb{R}^n$ with $n+1$ vertices and non-empty interior). Let ${\bf P} \in \mathbb{R}^{m \times n}, ...
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138
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Does there exist a function $f(X)$ with the following gradient $\mathrm{Tr}[(I-X)^{-1}]\cdot g(X)$?
Let $f: \mathbb{R}^{n\times n} \to \mathbb{R}$ be a function that receives a square matrix and spits out a scalar. Does there exist a function $f$ such that the gradient $\nabla_Xf(X) = \mathrm{Tr}[(...
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Decomposing a standard deviation [closed]
I am trying to "decompose" a standard deviation of an economy-wide variable into sectoral components. I have data for the year 2010 on the dispersion (standard deviation) of total economy ...
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How many matrices with given minors?
Let us consider a matrix $A \in M_{2 \times n}(\mathbb C)$ such that $rank A=2$. Let us denote by
$$
a_1,\ldots,a_d \in \mathbb C,
$$
where $d:=\binom{n}{2}$, the value of the $2 \times 2$ minors of $...
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Linear independence of complex polynomials and a "sum of squares" conjecture
This will take me some time to explain. Let $n \geq 2$ be a fixed integer. Let $p_i(z)$, for $i = 1,\ldots,n$ be $n$ nonzero complex polynomials of degree at most $n-1$. I am interested in ...
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Complexity of solving linear equations plus disequality constraints $a \ne b$
Let $K$ be ring and $S$ linear homogeneous system with $n$ variables $x_i$ over $K$.
Add to $K$ linear disequalities of the form $x_k \ne x_l$
and let the final system be $S'$. If $K=\mathbb{F}_2$,
$...
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Solution of complex linear system
In Brubeck, Nakatsukasa, and Trefethen - Vandermonde with Arnoldi (example 3) they solve the following linear system:
$$\operatorname{Re}\left(\begin{array}{ccc}1 & \cdots & z_{1}^{n} \\ 1 &...
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An otherwise linear matrix equation with the presence of a signum function : reference request
Consider the equation $$\pmb{c}+\text{sign}(G\pmb{c}) = L$$
$\pmb{c}$ is a $n\times1$ matrix.
$G$ is a $n\times n$ matrix which is also positive definite.
matrices $G$ and $c$ are real.
$L$ is a $n\...
- 698
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263
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Convergence of the eigenvector matrix for an analytic perturbation of a singular matrix
Let $A$ be an $n\times n$ matrix of all ones. Consider the analytic perturbation of $A$ as $$\tilde{A} = A + \epsilon H_1 + \epsilon^2 H_2 + \epsilon^3 H_3 + ... $$ All matrices are symmetric. Assume $...
- 698
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Entries of matrix iterates
We consider a matrix
$$A:=\begin{pmatrix} 0 & b & 0 &f \\a & 0 & e & 0 \\ 0 & d & 0 & h \\ c& 0 & g & 0 \end{pmatrix}.$$
This matrix has the interesting ...
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Are anti-linear maps/semi-linear, such as conjugations, linear in other almost complex structures?
I have asked this on mse, but I did not get any responses even after a bounty.
I started studying the book of Daniel Huybrechts, Complex Geometry An Introduction. I tried studying backwards as much ...
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Matrix iteration for non-negative matrices. Does it converge to some eigenvector?
Let $A$ be a non-negative (entrywise) matrix such that $A(1,1)>0$. Set $u=(1,0,0,...,0)^T$. Is it always true that there exists a non-negative eigenvector $v$ of $A$ such that $\lim_{n\rightarrow\...
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Proving equality of a vector multiplication example [closed]
I noticed that for any vectors $\mathbf{a},\mathbf{b},\mathbf{c}$ where $\mathbf{a},\mathbf{b}\in \mathbb{R}^{m\times 1}$ and $\mathbf{c}\in \mathbb{R}^{n\times 1}$, there exists the equality that
$$\...
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expectation of a quadratic function of a matrix variate normal distribution
I want to compute the following expectation term:
$E[{\bf{XA}}{{\bf{X}}^T}]$
where ${\bf X} \in R^{M \times M}$ and its elements are normal random variables such that
$vec\left( {\bf{X}} \right)\...
- 275
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Trace of a finite hypercubic tensor
Is the trace of a finite hypercubic tensor defined?
Clearly, for the bidimensional case $n \times n$ the trace is defined as the sum of the elements on the main diagonal:
$$\operatorname {tr} (\...
- 143
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Condition of tri-diagonal matrix A such that UAU^t=A for all orthogonal matrix U [closed]
$A,U \in \mathbb{R}^{n\times n}$
$\exists A \in \text{{tri-diagonal}} \quad s.t \quad UAU^{t}=A \quad \forall U \in \text{{orthogonal}}$
I know it holds when A is a diagonal matrix, but have no ...
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228
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Upper and lower bounds for a matrix norm with fixed diagonal
given a vertical vector $x=(x_1,x_2,...x_n)$ of size n and $B$ is a symmetric positive definite matrix ($n \times n$) of choice. All the diagonal elements of B are fixed to one, while extra diagonale ...
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Existence of rank-1 weight matrix in some type of deep neural network
Problem
This is the first time I have posted a question on this site and it may not be suitable for this venue, which is primarily used for research questions in maths. If someone finds it unsuitable,...
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For an one-order Linear Recurrence of a vector sequence, does the corresponding item follow a Linear Recurrence? [closed]
Consider an one-order Linear Recurrence of a vector sequence, such as
$${\bf x}_{n+1}={\bf A}{\bf x}_n$$
where ${\bf x}_n \in \mathbb{R}^m (\forall n)$, and ${\bf A} \in \mathbb{R}^{m\times m}$ and ${\...
- 601
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Solving Problem: LMIs and block matrices
I have been reading through this paper (https://ieeexplore.ieee.org/document/7995739) where I am stuck with this particular LMI. If you are familiar with control theory, the author is trying to find ...
- 103
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809
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Conjecture that relates matrix systems with some specific functions as solution sets
what is written below is a conjecture that I posed , and I ask for a proof or a disproof of it .I have checked the conjecture from $n$=$1$ up to $n$=$10$ using Matlab, and all results were in ...
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How eigenvalue perturbation affects back to the original matrix?
Both "eigenvalue perturbation" and "matrix perturbation" seems to study this scenario that given a matrix $A$, if we add something to it like $\tilde{A} = A+E$, how will the eigenvalues of $\tilde{A}$ ...
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Name of a matrix with one column and row removed [closed]
I am looking for the exact name of a matrix where the i-th column and rows have been removed.
I cannot remember how it is called in linear algebra, does anyone got an idea?
Thanks!
- 111
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root of identity matrix and lexicographic order
I asked a question here order of a permutation and lexicographic order but it seems*** that a very powerful and rich generalization can be made!
Let $A$ be a finite ring together with an arbitrary ...
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213
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How to decompose a matrix into its orthogonal and diagonal parts (assuming it has that form)? [closed]
Assume that $A = U * S$ for $U$ orthogonal and $S$ diagonal, ordered and positive.
If I only know $A$, is it possible to obtain $U$ and $S$?
My first guess would be taking the singular value ...
- 123
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Cholesky decomposition – non-positive definite matrix
In order to pass the Cholesky decomposition, I understand the matrix must be positive definite. However, I also see that there are issues sometimes when the eigenvalues become very small but negative ...
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138
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On sum of matrices
Suppose we have a matrix $M\in\Bbb Q^{n\times n}$ with no $0$ elements and we write as sum of two matrices $M_1$ and $M_2$ on following constraint.
$M_{1,ij}=M_{ij}$ and $M_{2,ij}=0$ or $M_{1,ij}=0$ ...
- 13.9k
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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 ...
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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
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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
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111
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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
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150
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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
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75
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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
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1
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110
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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
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327
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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
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1
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28
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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 ...
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