Questions tagged [linear-algebra]
Questions about the properties of vector spaces and linear transformations, including linear systems in general.
5,875 questions
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Seeking closed-form solution for vector equation
I'm working on a problem that involves vectors and scalar values, and I'm looking for a closed-form solution. I hope someone can help me with this or provide insights into how to approach it. Here's ...
- 111
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146
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Does the Plotkin bound mean that one can not achieve the Singleton bound asymptotically?
I am a little confused with the relationship between various bounds for error correcting codes.
Does the Plotkin bound mean that one can not achieve the Singleton bound asymptotically? That is, is ...
- 11
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163
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Lower bounds on eigenvalues of sum of two matrices (one of them is symmetric)
Let $L$ be a matrix with eigenvalues($\lambda$ $\geq$ 0). If I add a constant value (say $a$) to all the elements of $L$, what can we say about the minimum eigenvalue of this perturbed matrix?
Note: $...
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1
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93
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Positive linear recurrent sequence
Suppose there is a linearly recurrent sequence $a_k$ satisfies
$a_k\geq 0$ and $\sum_{k=1}^{\infty}a_k=1$.
Can we always find a $x$ and $r<1$ such that
$a_k\leq x r^k$?
- 1,503
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66
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Vector recurrences (asymptotic property)
Fix $m\in \mathbb{N}.$
For each $n\in \mathbb{N},$ let $A_n\in \mathbb{M}_{m}(\mathbb{C}),$ $X_n\in \mathbb{C}^m,$ and $B_n\in \mathbb{C}^m.$ Suppose that
$$X_{n+1}=A_n X_n+B_n,$$
$$\lim_{n\rightarrow ...
- 3
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2
answers
150
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Does $V \otimes V$ have non-trivial invariant subspaces under all unitaries of form $I \otimes U$?
Let $V$ be a finite-dimensional complex vector space, and $U$ be any unitary that acts upon it. We want to investigate interesting subspaces of $V \otimes V$ which are invariant under all unitaries of ...
- 766
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605
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Given a unitary $A$ and diagonal matrix $D$, find $B$ so that $AD=DB$
In principle the title says it all. Given some diagonal Matrix $D$ (which may have 0 and repeated entries) and some unitary matrix $A$, I want to find some other unitary matrix $B$ so that $AD=DB$. ...
- 101
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260
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Derivative involving a singular matrix
Find the first derivative of a SUM of all elements of an INVERSE of a square matrix (whose elements are functions of $z$) at $z=1$ knowing that all of the matrix' elements evaluate to $1$ at $z=1$.
...
- 419
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118
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Proving maximum value of a determinant of $I - B$, where $B$ is nonnegative matrix
I have the following setting:
Let $0 \leq r < 1$ and let $\{z_i\}_{i=1}^k$ be $k$ complex numbers such that $|z_i| \leq r$ for all $i$.
Moreover, $r + \sum_{i=1}^k 2Re(z_i) \geq 0$
I am interested ...
- 59
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263
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Change in the largest eigenvalue due to perturbation of diagonal components of a symmetric matrix
Let $A\in \mathbb{R^{n\times n}}$ be a symmetric negative difinite matrix and
$D\in \mathbb{R}^{n\times n}$ be a diagonal matrix $D = \mathrm{diag}\{d_i\}, (d_i < 0)$.
From Weyl's inequality, the ...
- 1
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168
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Find a matrix transformation for 2 vectors with conditions
If we know that there are two vectors $x,y\in\mathbb{R}^d$ satisfying
\begin{equation}
\|x\|\ge c_1 \|y\|, \quad x^Ty\ge c_2\|x\|^2,
\end{equation}
where $c_1>0$ and $c_2>0$ are some given ...
- 97
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Standard Gram matrices for lattices
I would like to define standard Gram matrices, and use them to help me understand the symmetries of lattices.
I define "standard Gram matrix" as the Gram matrix g that minimizes the ...
- 53
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199
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Maximum dimension of a simultaneous anisotropic subspace of quadratic forms over $ \mathbb{Q} $
Let $(V,q )$ be a quadratic space over $ \mathbb{Q} $. A subspace $ U $ is called totally isotropic if $ q(x) = 0 $ for all $ x \in U $ and a subspace $ U $ is called an anisotropic subspace if $ ...
- 923
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Does binary extension field multiplication matrix have random rows?
Any element in a boolean extension field $a\in GF(2^n)$ can be presented by a boolean vector $a_{(2)} \in GF(2)^n$. For any element $a\in GF(2^n)$, there exists a boolean matrix $M_a\in GF(2)^{n\...
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For which $\beta \in \mathbb{F}_{p^k}$, $\{1,\beta,\beta^2,\cdots,\beta^{k-1}\}$ form a basis of $\mathbb{F}_{p^k}$? [closed]
Let $p$ be a prime, and let $\mathbb{F}_{p^k}$ be a finite field. For which $\beta \in \mathbb{F}_{p^k}$, $\mathcal{B}_{\beta}:=\{1,\beta,\beta^2,\cdots,\beta^{k-1}\}$ form a $\mathbb{F}_p$-basis of $\...
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157
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Can I rewrite this expression as a Kronecker product?
Let $M$ be a symmetric positive definite matrix, and let $a > 0$ be a given constant.
Let $\text{vec}(\cdot)$ denote the operator that stacks vertically the columns of a $d \times d$ matrix into a $...
- 219
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Does positivity of the n(n-1)/2 principal minors formed from 2 x 2 submatrices ensure positive-definiteness of the n x n matrix itself?
I am interested in conditions under which an $n \times n$ matrix ($\rho$) is positive definite. Of course, one necessary and sufficient set of conditions is that the $n$ leading minors of $\rho$ each ...
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Define an inner product between p-blades so that 0= completely orthogonal and 1=completely overlapping for their subspaces
$\DeclareMathOperator\span{span}$Denote two $p$-blades $\nu=v_1\wedge \dots \wedge v_p$ and $\omega=w_1\wedge \dots \wedge w_p$ $\in \bigwedge^p X$, where $X$ is an inner product space. How to define ...
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Generalization of Dickson's Lemma
Given $\{v^i\}_{i \in \mathbb{N}} \subseteq \mathbb{N}^n$, and $\cup_{k=1, \ldots, m} C_j = \mathbb{N}^n$ for some $m$, where each $C_k$ is a cone generated by rational vectors. My question is: does ...
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215
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Principal minors and similarity
Given two real and irreducible matrices $A$ and $B$ of size $n \times n$. A matrix $A$ is irreducible if there is no permutation matrix $Q$ so that
$$
Q^{-1} A Q = \begin{bmatrix} E & G \\ 0 & ...
- 909
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1
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74
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Conditions for solution to equation [closed]
I have the following question in an exercise of linear fitting.
Q.
We have $n$ linearly independent known vectors $\textbf{a}_i$, $i = 1, . . . , n$, a known vector $\textbf{b}$, and an unknown vector ...
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1
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78
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Similar to inverse plus rank 1
Given a real, invertible matrix $A$. For which vectors $b$ and $c$ is
$$
A^{-1} + bc^T
$$
similar to $A$? And is the rank-1 matrix $bc^T$ unique?
- 909
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115
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The product of non-primitive matrices with zero positions in common
Def.[1]:
A non-negative $n \times n$ matrix $A$ is called a non-primitive if there is no an integer $k$ such that all entries of $A^k$ are positive.[1]
Def.[2]:
Let ${\bf A}=(a_{i,j})$ and ${\bf B}...
- 313
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1
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91
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Choosing the best submatrix
Let $\mathbf{A}_{m\times n}$ be a matrix with non-negative elements. Assume that a submatrix $\mathbf{B}$ from $\mathbf{A}$ is defined as
\begin{align}
B_{i,j} =
\begin{cases}
A_{i,j}, & i\in\...
- 287
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154
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Energy of a symmetric matrix with $0$, $1$ or $-1$ entries
I have a symmetric matrix with entries $0$, $1$ or $-1$ which appeared in my works in graph theory (the diagonal entries are all zero). I need a good upper bound for the energy of this matrix; i.e. "...
- 351
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224
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Positive definite matrix
We have $a_1,a_2,...,a_n\in (0,1)$ and matrix
M=
\begin{bmatrix}2a_1&a_2&a_3&.&.\\a_2&2a_2&a_3&.&.\\a_3&a_3&2a_3&.&.\\.&.&.&.&.\end{...
- 163
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320
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Expectation of a linear operator
We define $T: C[0,1]\to C[0,1]\ni T(f(x))= \sum\limits_{k=1}^{m} p_k (f\circ f_k)(x):=\mathbb E( f(X_{n+1}|X_n=x)$ for a system $X_{n+1}=f_{\omega_n}(X_n), n=0,1,2\dots.$ and $\omega_n$ are i.i.d ...
- 149
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115
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Infinite norm of two randomly picked points [closed]
Let X and Y be points in the 4000-dimensional unit cube, picked at random with uniform distribution, which means from I what I understand that all locations in the cube are equally likely. $X \in [0,1]...
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Pre-garbling does not improve capacity of a channel
Suppose ABC=B for some column stochastic matrices A, B, and C.
Can the following implication be made without further restrictions:
There necessarily exists a column stochastic matrix D such that DB=BC?...
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140
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About roots of the equation $det(A-\lambda B)=0$ [closed]
Let $A,B$ be square real symmetric matrics of the same degree $n \geq 3$ without common isotropics vectors (i.e. there is no a nonzero vector $x\in \mathbb R^n$ such that $x^TAx=x^TBx=0$).
Are roots ...
- 9
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109
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prove the singularity of a matrix as solution of a non-linear equation
Let $B$ ($n \times n$) and $R$ ($m \times m$) be two square matrix with $n>m>0$ who satisfie:
$B=(I-KH)B(I-KH)^T+K RK^T$
with $K=BH^T(HBH^T+R)^{-1}$ and $rank(H)=m$
I would like to prove $...
- 159
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215
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Vanishing bilinear forms
For a symmetric or antisymmetric bilinear form $\varphi$ on a vector space $V$, if $\varphi(x,y)=0$ then also $\varphi(y,x)=0$ ($x,y\in V$).
I was wondering if this is also a necessary condition for ...
- 25
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269
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Limit of eigenvalues of a matrix perturbation sequence
Suppose $H$ is an $n\times n$ symmetric positive definite matrix, $M_k$ is a sequence of $n \times n$ matrix (not necessarily symmetric) such that $M_k \to O$ where $O$ is the zero matrix. Let $\...
- 135
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Kernel vectors with given number of non-zero entries?
Let $A$ be a $n\times n$ real matrix. Is there a vector $\vec x \in \mathbb{R}^n$ with exactly $0 \le k < n$ zero entries such that $A \vec x = 0$?
Is there an efficient algorithm to tackle this ...
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Does stability imply that the logarithmic norm is negative?
Let $A \in \mathbb R^{n\times n}$ and assume that all eigenvalues lie in the left open halfplane.
Is it true that the logarithmic norm $\mu_2 (A):= \lambda_{\max}
\left(\frac{A + A^T}{2}\right)<0?...
user74165
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On the spectrum of Hermitian matrices [closed]
I'm working on the adjacency matrix of some graphs and need some facts about Hermitian matrices which have exactly two distinct eigenvalues. Can anybody help me introduce source about spectrum of ...
- 351
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258
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Rank of the connected components
Let $G$ be a simple graph with adjacency matrix $A(G)$. Let $v$ be a cut-vertex of $G$. Let $G_1, G_2,\dots, G_k$ be the connected components of the induced graph $G-v$ ( the subgraph resulting after ...
- 640
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390
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An upper bound for skew symmetric rank 2 matrices
Earlier, I had asked a similar question but that was not the correct problem where I got stuck. After a few quick answer, I realized that and I apologize for that.
Let $B_m$ be the space of all skew-...
- 179
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1
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553
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Directed graph cycles and the inverse of a weighted adjacency matrix
Let us view a matrix $B \in \mathbb{R}^{n \times n}$ as the weighted adjacency matrix of a directed graph $G$, i.e. there is an edge $i \to j$ in $G$ if $B_{ij} \neq 0$. Assume further that
$B$ does ...
- 1,731
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2
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160
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A matrix between vectors, and inequality!
I have an inequality as follows
$$s^T\phi\leq -|s|^TA$$
where $s$, $\phi$ and $A$ are vectors with appropriate dimensions. I want to prove that this inequality holds for the following too
$$s^TM\...
- 29
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69
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Can there be underdetermined linear systems whose set of minimal support solutions is infinite?
Let $A \in \mathbb{F}^{m \times n},$ $m<n,$ and let $b \in \mathbb{F}^{m \times n}$ be such that the system $Ax=b$ is consistent. Does it follow that the set $X$ of minimal support solutions of ...
- 259
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397
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determinant of the hadamard inverse of a positive matrix
Let $A=(a_{ij})_{i,j=1}^n$ be a positive definite matrix with $a_{ij}>0$ for all $i, j$. Define the Hadamard inverse $A^{\circ -1}$ of $A$ as $(a_{ij}^{-1})_{i,j=1}^n$. Is it possible to decide ...
- 41
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122
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Nilpotent infinite binary matrices
Let $\text{Mat}(\mathbb{N},\{0,1\})$ be the set of all maps $A:\mathbb{N}\times\mathbb{N}\to \{0,1\}$. We define a matrix multiplication for $A, B\in \text{Mat}(\mathbb{N},\{0,1\}$) and $m,n\in\mathbb{...
- 48.9k
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350
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The order of companion matrix over various modulo
We consider a positive integer number and call it our modulo and denote it with $m$. We choose a positive integer number like $p$ and
call it the degree of our polynomial. We select $p$ integer ...
- 313
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votes
1
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184
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Jordan Decomposition for a certain matrix
This is the form of matrices I am concerned about recently,
Denote by $A=(a_{ij})_{e \times e}$, where $a_{ij}$ is block matrix of dimension $n_i \times n_j$. Now given $n, 0 \leq n \leq e-1 $. Let $...
- 363
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1
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311
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Intuitive explanation of concentration of the measure for spheres [duplicate]
What is the concentration of the measure(c.o.m.)?
I am struggling with the following sentence;
"The phenomenon of the concentration of the measure for spheres in dimensions larger than 2."
I tried ...
- 11
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votes
1
answer
524
views
Compose/decompose rotation matrix from/to plane of rotation and angle
I would like to compose/decompose an $n$-dimensional orthogonal rotation matrix (restricting to simple planar rotations, which rotates in the specified plane of rotation, and fixes in the plane ...
- 111
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1
answer
720
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Unique solution to a matrix equations [closed]
Given any $n \times k$ real matrix $M$, where $n<k$ and $rank(M)=n$, I consider the following equation (where $M'$ is the transpose of $M$):
$$
MM' = MAM'
$$
Then clearly, $A = \mathbb{1}_k $, ...
0
votes
1
answer
317
views
Some questions related to the unitary operators
A unitary operator is a surjective linear operator between complex inner product spaces, which preserves the inner product.
What is the name of the analogue for the real case? Orthogonal operator ...
- 5,529
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1
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100
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How does the rank of $C_i$ change with $i$?
Let $k$ be a field. Let $A,B\in k^{m\times n}$ and $$C_i=\pmatrix{A&B&&&\\&A&B&&\\&&\ddots&\ddots&\\&&&A&B}\in k^{im\times(i+1)n}.$$ ...
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