Questions tagged [linear-algebra]
Questions about the properties of vector spaces and linear transformations, including linear systems in general.
6,058 questions
8
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Eigenvector quadratic form inequality
Given matrix $B\in\mathbb{R}_+^{n\times n}$ and scalar $\alpha \in \mathbb{R}_{+}$, let $A:=\alpha B+B^T/\alpha$. Note that $B$ and $A$ have nonnegative entries and that $\alpha$ controls degree of ...
- 535
0
votes
3
answers
263
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Eigenvalues of block matrix with $A$ on diagonal blocks and $B$ on off-diagonal blocks [duplicate]
Let $A, B \in {\Bbb R}^{2 \times 2}$ be two self-adjoint matrices. I am interested in the following block matrix
$$ M =
\begin{bmatrix}
A & B & B & \dots & B \\
B & A & B & ...
1
vote
3
answers
395
views
Binomial coefficient C(2k,n-1) alternative formula equivalent to the Vandermonde identity?
I am working with a binomial sum that arises in some combinatorial arguments (and also appears in certain generating‐function manipulations). Specifically, I have this identity
$$
\sum_{j=0}^{n}
\...
- 13
3
votes
1
answer
179
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Effect of row permutations on the Perron vector
If $A$ is a positive matrix with Perron root $\rho(A)$ and Perron vector $v$ (strictly positive), and $P$ is a permutation matrix, how does the Perron vector and root of $B = P A$ relate to that of $A$...
- 65
1
vote
1
answer
153
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Dimension of the set of all separable mixed states
Let $\mathcal{H}_1$ and $\mathcal{H}_2$ be two finite dimensional Hilbert spaces, say of complex dimensions $m$ and $n$. One can ask about the (real) dimension of various sets of states, for example ...
- 9,634
3
votes
1
answer
212
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Least-squares problem with a rank constraint
Given $n < m < n^2$, fat rank-$m$ matrix ${\bf A} \in {\Bbb R}^{m \times n^2}$ (that has full row rank) and vector ${\bf y} \in {\Bbb R}^m$,
$$\begin{align}
\underset{{\bf X} \in {\Bbb R}^{n \...
- 73
7
votes
1
answer
353
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Estimate the Hilbert–Schmidt norm of $\mathrm e^{\mathrm itA}-\mathrm e^{\mathrm itB}$
Let $A$ and $B$ be real-valued self-adjoint $(n \times n)$-matrices with $\|A\|,\|B\| \leq 1$, where $\|\cdot\|$ is the operator norm in the $\ell^2$ sense. Assume that $D:=A-B$ is a $0$-$1$ matrix ...
- 699
9
votes
1
answer
457
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Localization of eigenvalues on complex plane
Let $B$ be a cyclic upper-triangular nonnegative matrix,
$$B = \begin{bmatrix} 0 & b_1 & 0& \dots &\dots &0 \\
0 & 0 & b_2 & 0& \dots & 0\\
\vdots &\vdots&...
- 535
0
votes
0
answers
80
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Computational complexity of a system of linear diophantine equations with a non zero constraint
Given a system of linear diophanthine equations. What is the computational complexity of checking if the system has a solution or not or finding a solution if we have an additional constraint that ...
- 215
5
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1
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431
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A (new?) matrix composition operation and its compositional laws
$\DeclareMathOperator\perm{perm}$Let $A$ be an $n \times n$ matrix with arbitrary complex entries and $1 \le i \le n$ an index and let $A^i$ the the matrix $A$ with the $ith$ row and column deleted.
...
- 2,109
3
votes
0
answers
282
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Matrices over $\mathbb{Z}_l$
Suppose I have a matrix over the $l$-adic integers $\mathbb{Z}_l$ which is diagonalizable over $\mathbb{Q}_l$. How to classify such matrices by similarity over $\mathbb{Z}_l$?
- 1,024
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votes
0
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74
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Existence of solutions for system of quadratic equations
I found an interesting problem in Post 1, Post 2. Let us suppose to have $ M $ quadratic equations
$$
\underline{x}^T A_i \underline{x} + \underline{b}^T \underline{x} = c \quad i = 1,...,M
$$
with $ \...
- 23
14
votes
1
answer
1k
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What is the determinant of a set?
Cross-posted from MSE where the question didn't get much attention. The question is related to and has a similar motivation as this MO question.
Let$\newcommand{\from}{\colon}\newcommand{\sgn}{\...
- 1,628
10
votes
2
answers
660
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Many linearly independent functions without choice
Do there exist $2^{|\mathbb R|}$ linearly independent functions
$\mathbb R\to\mathbb R$ in Zermelo-Fraenkel set theory (without choice)?
I guess that constructing an explicit large linearly ...
- 4,236
1
vote
1
answer
207
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Action of profinite powers of an integer matrix in $\operatorname{SL}(2,\mathbb{Z}_p)$ on $\mathbb{Z}^2$
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\tr{tr}$Let $A\in \SL(2,\mathbb{Z})$. Inside $\SL(2,\mathbb{Z}_p)$, we can consider profinite powers $A^r$, where $r\in\widehat{\mathbb{Z}}$. For ...
- 7,785
1
vote
0
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284
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problem with a calculation in Biomathematics paper
I'm working on this paper but really struggling to understand how the derived equation (10). It does seems like something is missing or its abscense it's not well justified. Can someone help me?
link ...
- 191
0
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0
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111
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Diagonal properties for matrices $( I + \frac{n}{n+1} X)^{-1}$ with $X \succeq 0$
Let $X \in \mathbb{R}^{n\times n}$ be a symmetric positive semidefinite matrix. We define
$$ A = \left( I + \frac{n}{n+1} X \right)^{-1}, $$
and set $B= A(X + I)$ and $C=AX$. My goal is to demonstrate ...
- 2,602
2
votes
1
answer
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Deciding Total Unimodularity of Circulant Matrices
Question:
which, if any, of the criteria for total unimodularity of matrices are easier to check, in the sense of computational complexity, for a circulant matrix $A\in\lbrace -1,0,+1\rbrace^{n\times ...
- 14k
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0
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81
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Least-square distance between an array of quadratic forms and a given positive vector
Suppose we are given a list of $N$ positive definite quadratic forms $X^TQ_k X$ (where $k\in[1,N]$ and
$Q_k\in\mathbb{R}^{p\times p}$ $\forall k$), and a positive vector $V$ of same length $N$ i.e. $V=...
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1
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0
answers
152
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An invertibility property of submatrices of a matrix
Let $\mathbb{F}_n$ denote the finite field with $n$ elements.
Suppose that the (non-tall) matrix ${\bf M} \in \mathbb{F}_n^{r \times n}$, where $r \leq n$, has rank $r$ and for any $k \leq r$, the ...
- 11
2
votes
0
answers
67
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Computing determinants of endomorphisms of Drinfeld Modules
Reposting from MathStackExchange https://math.stackexchange.com/questions/5028883/determinant-of-an-endomorphism-of-a-drinfeld-module-over-a-finite-field with a slightly more general question.
Let $\...
- 191
1
vote
1
answer
436
views
Show $2\times 2$ matrix has positive eigenvalues
Let the real matrix $\bf A$ be positive definite and let $c>0$. Consider the matrix
$${\bf M} := \begin{bmatrix}
0 & -\text{A}_{11}-\frac{c^2}{4} & \frac{c}{2} & -\text{A}_{12} \\
1 &...
8
votes
1
answer
422
views
Almost disjoint family of vector subspaces
Let $\mathbb{F}$ be a countable field, and let $E$ be an $\mathbb{F}$-vector space of dimension $\aleph_0$. If $U,W \subseteq E$ are infinite-dimensional subspaces of $E$, we say that $U$ are $W$ are ...
- 1,713
1
vote
0
answers
141
views
Wronskian, linear dependence, and multi-variables
The Wronskian is a useful and standard object for testing the linear dependence of a set of differentiable functions ${y_i (x)}$. Are there similar useful ways for testing the linear dependence of a ...
- 149
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0
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169
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Complex traces in positive characteristic — is there a deeper meaning?
Let $\mathbf{A}\subseteq\mathbf{C}$ be the full ring of algebraic integers, and let $M<\mathbf{A}$ be a maximal ideal containing $p$. Then $k=\mathbf{A}/M$ is (isomorphic to) the algebraic closure ...
- 2,393
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0
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45
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Diagonal dominance of matrices $M = (I + XY)^{-1}X$ with $X$ symmetric positive semidefinite and $Y$ Laplacian
We denote by $\mathcal{M}_n$ the space of $n \times n$ matrices $X$ such that, for any Laplacian matrix $Y$ (i.e. $Y$ is symmetric, has non-negative diagonal coefficients, non-positive off-diagonal ...
- 2,602
2
votes
0
answers
175
views
Estimation for solution of the Lyapunov equation with semidefinite right-hand side
Consider the Lyapunov matrix equation in symmetric matrix unknown $\bf X$
$$ {\bf A}^\top {\bf X} + {\bf X} {\bf A} = − {\bf B} {\bf B}^\top$$
where the matrix $\bf A$ is Hurwitz. We know that its ...
- 21
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votes
0
answers
64
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Diagonal dominance of $(\mathrm{Id} + XY)^{-1} X$ with $X$ block diagonal and $Y$ Laplacian
$\newcommand\Id{\mathrm{Id}}$Let $X \in \mathbb{R}^{n \times n}$ be the block diagonal matrix
\begin{equation}
X =
\begin{bmatrix}
J_1 & 0 & \cdots & 0 \\
0 & J_2 & \cdots & 0 ...
- 2,602
2
votes
0
answers
102
views
Centraliser algebra of a poset
Let $P$ be a finite poset with $n$ elements and $C_P$ the Cartan matrix of $P$, that is the $n \times n$-matrix with entries $c_{i,j}=1$ if $i \leq j$ and zero else.
Let $K$ be a field of ...
- 28.1k
2
votes
1
answer
365
views
Best notation for tensor product with associativity
I want to express the product of a three-dimensional array by two one-dimensional vectors over some ring $R$:
$$r = A \cdot b \otimes c$$
where
$A \in R^{\ell \times m \times n}$
$b \in R^n$
$c \in R^...
- 147
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0
answers
133
views
How to prove the solution of system of random equations concentrates on the solution of noiseless case?
Consider a set of unit vectors $\left\{ {\bf v}_1, {\bf v}_2, \dots, {\bf v}_n \right\} \subset {\Bbb R}^2$ (that are not all in parallel). Let $\bf A$ be the adjacency matrix of a complete graph with ...
- 149
0
votes
0
answers
90
views
Bounds on the eigenvalues of perturbations of a symmetric matrix
This question was previously posted on MSE.
Let us fix $\varepsilon\in (0,1)$ and $\beta\in\mathbb R$.
Consider the $2 n\times 2n$ symmetric tridiagonal probability matrix
$$Q_n :=\begin{bmatrix}
1-\...
- 345
3
votes
1
answer
194
views
Projective linear group $\mathrm{PGL}(2,p)$ with elements of order $p+1$
It is clear from the character table of $\mathrm{PGL}(2,p)$ that there exists an element $g \in \mathrm{PGL}(2,p)$ such that the order of $g$ divides $p+1$. (Take $g = \bigl( \begin{smallmatrix}1 &...
6
votes
0
answers
267
views
What assumptions are required of an addition operator to make it the standard vector addition on Euclidean space?
Suppose we have an operation $\oplus$ on $\mathbb R^2$ satisfying:
$\oplus$ together with the standard scalar multiplication forms a vector space structure on $\mathbb R^2$.
$\oplus$ is equivariant ...
- 693
2
votes
2
answers
508
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Evaluating derivatives of a polynomial by evaluating the polynomial at multiple points
Let's consider $\mathbb{R}[X]$ the vector space of real polynomials with one indeterminate and $\mathbb{R}[X]_n$ the sub-vector space of polynomials with degree at most equal to $n$. For all possible $...
- 259
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votes
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Two SPD matrices are identical?
Two SPD matrices admits eigen-decomposition $\Sigma_p=U_p S_p U_p^{\top}$ and $\Sigma_q=U_q S_q U_q^{\top}$, where $S_p$ and $S_q$ contain ordered eigenvalues that are distinct. Let $\Sigma_v=U_q^{\...
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votes
1
answer
365
views
Standard form for additive group in $\mathrm{GL}_n$
$\DeclareMathOperator\GL{GL}$Let $G$ be a closed subgroup of $\GL_n(K)$, where $K$ is an algebraically closed field, and assume that $G$ is isomorphic to the additive group $\mathbb{G}_a$, or ...
- 7,946
0
votes
1
answer
161
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Let $A$ be a SPD matrix. Suppose diagonal $(A)_{ii}$ equals to its eigenvalue $\lambda_i$. Must $A$ be a diagonal matrix?
Given a symmetric positive definite (SPD) matrix with eigendecomposition $A = U \Lambda U^{\top}$ follows $\operatorname{diag}(A)=\operatorname{diag}(\Lambda)$, is it necessary to have $U=I$ so that $...
- 21
3
votes
1
answer
345
views
When can we cancel in direct sum (or direct product) isomorphisms for algebraic structures?
In the context of finite groups and vector spaces, cancellation in direct sum (or direct product) isomorphisms is well-understood: i.e for finite dimensional vector space if $V \oplus W \cong V \oplus ...
- 359
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0
answers
124
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Maximal commutative subalgebras of $\mathfrak{sl}(5)$
I am looking for a list of maximal commutative subalgebras of $\mathfrak{sl}(5)$.
Suprunenko, D. A., & Tyshkevich, R. I. (1968). Commutative matrices.
Give a list starting on p132, however, I ...
- 241
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votes
1
answer
431
views
Is 3-space the disjoint union of affine lines, one in each direction?
For each 1-dimensional subspace L of ℝ3, let A(L) be some affine line parallel to L.
Is there an assignment A with the properties that a) all the A(L) are disjoint and b) the union of all the A(L) is ...
- 3,555
0
votes
0
answers
110
views
Determining if the two different dimension integer matrices are congruent
Given a matrix $A\in M_{6\times 6}(\mathbb{Z})$ that is symmetric and has determinant zero. I want to deterministically figure out if there exists a matrix $T\in M_{6\times 5}(\mathbb{Z})$ such that $...
- 514
2
votes
0
answers
230
views
For which $X$ has the matrix $M = (I + X^T X A)^{-1} X^T X$ dominant diagonal coefficients?
Let $A \in \mathbb{R}^{n \times n}$ be a Laplacian matrix (i.e. symmetric such that $A_{ii} = - \sum_{j \neq i} A_{ij} > 0$ for any $i$ and $A_{ij} \leq 0$ for any $i \neq j$), and $X \in \mathbb{R}...
- 2,602
2
votes
2
answers
332
views
Property of some positive definite matrix and positivity problem
Consider a matrix $X \in \mathbb{R}^{m \times n}$ whose colums are $x_1, \ldots , x_n \in \mathbb{R}^m$ and set
$$M = Id + X^T X (n Id - J) \in \mathbb{R}^{n\times n}$$
where $J \in \mathbb{R}^{n \...
- 2,602
5
votes
1
answer
469
views
Spectral norm of matrix when we change each entry to have positive sign
Let $A$ be a positive semi-definite $(d \times d)$-matrix with real entries. Let $B = (|a_{i,j}|)_{i,j \leq d}$ be the similar matrix, where each entry is non-negative. Let $||A||$ and $||B||$ denote ...
- 1,389
5
votes
3
answers
714
views
Symmetric product of $\mathbb{P}^{1}$
Let $S_{n}$ be the $n$-th symmetric group. I want to know why the symmetric product
$$(\mathbb{P}^{1})^{(n)}=(\mathbb{P}^{1}\times\cdots\times\mathbb{P}^{1})/S_{n}$$
is biholomorphic to $\mathbb{P}^{n}...
- 69
3
votes
0
answers
154
views
Simultaneous left-right orbit of a set of matrices
Let $A_1,\dots, A_k$ be non-zero $n \times n$ complex matrices, and let $1 \leq r \leq n$ be an integer. I want to know if there exists a polynomial time algorithm to decide if there exist $r \times n$...
- 970
3
votes
2
answers
234
views
A variant of the vertex clique cover problem
Assume that I have a graph $G(V, E)$, where each vertex $v$ is assigned a non-negative weight $w(v) \geq 0$. I aim to find a partition of the graph into cliques $C_1, C_2, \ldots, C_k$ (Here, $k$ is ...
- 199
0
votes
0
answers
118
views
Solving 2nd order partial differential equations
I have a study that includes the following system of 2nd order partial differential equations. The unknown variables in these equations are $y(k,t)$, $x(k,t)$, $z(k,t)$, and their derivatives.
My ...
- 117
1
vote
0
answers
137
views
transforming coordinates between cartesian and spherical [closed]
I apologize if this has been asked before. I searched on this site and others but didn't find this particular issue. My background is physics and math.
My question is near the end of the post. First, ...
- 11