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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Near-linear mappings from $\mathbb F_p$ to $\mathbb R$
$\newcommand{\F}{{\mathbb F}}$
$\newcommand{\R}{{\mathbb R}}$
$\renewcommand{\phi}{\varphi}$
Let $p\ge 5$ be a prime.
If the functions $\phi_1,\phi_2,\phi_3\colon\F_p\to\R$ satisfy $\phi_1(x)+\...
- 23k
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Seeking proof for linear algebra constraint problem.
Given a symmetric real matrix with a zero diagonal $M$, I am trying to find a diagonal matrix $D$, such that the matrix $M + D$ is positive definite, and $(M+D)^{-1}$ has a diagonal consisting of all ...
- 379
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Left and right eigenvalues
A quaternionic matrix $A$ gives rise to a
function $\mathbb{H}^n \to \mathbb{H}^n$
given by $x \mapsto A \cdot x$. This is real linear,
but not complex- or quaternionic-linear
(in general) if we ...
- 12.5k
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A difficult determinant
(EDIT: I have removed the denominators I had in a previous version as they were superfluous)
The $N\times N$ determinant
$$D(a,\vec{b})=\det\left((2N+a+b_j-i-j)!\right)$$
has the nice form
$$D(a,\...
- 2,552
13
votes
1
answer
889
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Probability that random nonnegative integer matrix is singular
Q. What is the probability that an $n \times n$ matrix, whose elements
are independent uniformly random integers in $\{0,1,\ldots,k\}$, is singular?
For example, for $n=3$ and $k=2$, the first ...
- 151k
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answer
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When is a matrix similar to a non-negative matrix?
Consider a real square matrix $A$ of size $n\times n$. Under which conditions on $A$ does there exist a row-stochastic matrix $U$ (non-negative, rowsums = 1), such that $A'=U^{-1}AU$ is a non-negative ...
- 231
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Spectral properties of finite metric sets
Given a finite metric set $S=\{P_1,\dots,P_n\}$, one gets a real symmetric matrix $M=M(S)$
with rows and columns indexed by elements of $S$ by setting
$M_{i,j}=d(P_i,P_j)$.
It is easy to see that $M$...
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Determinant of $V^* V$ where $V$ is rectangular Vandermonde matrix with nodes on unit circle
Let $z_{1},\dots,z_{k}$
be distinct complex numbers with $\left|z_{j}\right|=1,\;j=1,\dots,k$. For any natural $N\geqslant k$
consider the rectangular Vandermonde matrix
$$
V_{N}=\begin{pmatrix}1 &...
- 959
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A generalization of the Powers-Stormer inequality
The well-known Powers-Stormer inequality says the following: for positive semidefinite operators $A, B$, we have that $\mathrm{Tr}((A - B)(A - B)) \leq \| A^2 - B^2 \|_1$, where $\| \cdot \|_1$ ...
- 2,019
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Eigenvalues of submatrices
I am interested in results on the eigenvalues of submatrices.
Given a symmetric and positive-semidefinite matrix $M$, denote the submatrix obtained by deleting the ith column and jth row as $[M]_{ji}$...
- 599
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2
answers
795
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Distance of vectors versus distance of their difference vectors
For any given $x \in \mathbb{R}^n$, let $\nabla{x} \in \mathbb{R}^{n \choose 2}$ be the vector whose $\{i,j\}$-th entry is $|x_i-x_j|$. I think the following claim is true.
Claim. If $f, g \in \...
- 519
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Unit polynomial vector fields on the sphere
Let $\mathbb{S}^3 \subset \mathbb{R}^4$ be the unit $3$-sphere. Is there a classification available for $3$-homogeneous polynomial, unit norm, vector fields on $\mathbb{S}^3$?
More explicitly, a $3$-...
- 501
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Seek for a algebro-geometric proof: the group homomorphism $\mathrm{SL}(2,\mathbb{Z}) \rightarrow \mathrm{SL}(2,\mathbb{Z}/N\mathbb{Z})$ is surjective
It is a well-known fact that the group homomorphism $\mathrm{SL}(2,\mathbb{Z}) \rightarrow \mathrm{SL}(2,\mathbb{Z}/N\mathbb{Z})$ is surjective.
What I want is a proof by method of algebraic geometry. ...
- 1,168
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A Dynkin type classification result in linear algebra
Let $G$ be a finite directed acyclic graph. The Cartan matrix $C_G=C$ of $G$ is defined as the matrix with rows and colums indexed by the vertices of $G$ and $c_{i,j}$ counts the number of paths from $...
- 26.5k
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Can one Gershgorin circle (only) contain all eigenvalues, when the other circles are not contained in it
In short, following a question from my students, I am trying to find a special case where all the eigenvalues of a matrix lie within only one circle, but not in the others, and the other circles are ...
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Is the set of power matrices decidable?
Let $\text{Mat}(n\times n,\mathbb{Z})$ denote the collection of integer $n\times n$ matrices. We say $M\in \text{Mat}(n\times n,\mathbb{Z})$ is a power matrix if there is an integer $k>1$ and a ...
- 48.9k
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A determinant problem for primes $p\equiv 1\pmod4$
Let $p$ be an odd prime, and let $A_p$ denote the matrix
$$[a_{ij}]_{1\le i,j\le (p-1)/2},$$
where
$$a_{1j}=\left(\frac jp\right),\ \ \text{and}\ \ a_{ij}=\left(\frac{i^2+j^2}p\right)\ \text{for}\ i&...
- 15.6k
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Pointwise (Hadamard) matrix product and the rank
$\DeclareMathOperator{\rk}{rk}$
Suppose that $A$ is a square matrix of order $n$. If, for any polynomials $P$ and $Q$ with $\deg P+\deg Q\le 2$, we have
$$ P(A)\circ Q(A^t) = P(1)Q(1)\, I_n \tag{$\...
- 23k
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Regular languages of matrices and their generating functions
My question is somewhat related to this question.
Let us fix natural numbers $k$ and $C$. Let $A$ be an automaton whose alphabet consists of $k\times k$ matrices with integer coefficients of ...
- 3,897
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6
answers
2k
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Differentiability of eigenvalues of positive-definite symmetric matrices
Let $A\in M(n,\mathbb{R})$ be an invertible matrix. Consider the (real) eigenvalues $\lambda_1,\cdots,\lambda_n$, in increasing order, of the positive-definite symmetric matrix $A^t A$. We shall ...
- 3,423
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answers
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Is any $(n-1)\times (n-1)$ submatrix of an $n \times n$ Vandermonde matrix invertible?
Given an $n \times n$ vandermonde matrix $V$ which is invertible, is any $(n-1) \times (n-1)$ submatrix of $V$ invertible also?
I think the answer is yes, but I don't know how to prove.
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How can I learn about doing linear algebra with trace diagrams?
There is a wikipedia article. There is a paper by Elisha Peterson. I tried reading these but they don't seem to click for me.
Are there books or other resources for learning how to do linear algebra ...
- 3,613
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6
answers
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Is there an analog of determinant for linear operators in infinite dimensions as that of finite dimensions?
I am trying to find out the essence of what a determinant is. Besides, in finite dimensions, determinant is the kind of numerical invariant that determines the invertibility of a linear operator, but ...
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Kind of submultiplicativity of the Frobenius norm: $\|AB\|_F \leq \|A\|_2\|B\|_F$?
Let $\|\cdot\|_F$ and $\|\cdot\|_2$ be the Frobenius norm and the spectral norm, respectively.
I'm reading Ji-Guang Sun's paper 'Perturbation Bounds for the Cholesky and QR Factorizations' from BIT ...
12
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3
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Can the equation $1+z+z^2=z^n$ for natural $n$ have multiple complex roots $z$?
The question is stated in the title of this post.
It is easy to see that, if $z$ is a multiple root of $p_n(z):=1+z+z^2-z^n$, then $(n-2)z^2+(n-1)z+n=0$, so that we can successively express $z^2,\dots,...
- 128k
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The character table of the symmetric group modulo m
Let $S_n$ be the symmetric group and $M_n$ the character table of $S_n$ as a matrix (in some order) for $n \geq 2$.
Question: Is it true that the rank of $M_n$ as a matrix modulo $m$ for $m \geq 2$ ...
- 26.5k
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Matrix inequality $(A-B)^2 \leq c (A+B)^2$ ?
Let A and B be positive semidefinite matrices. It is not hard to see that $(A-B)^2 \leq 2A^2 + 2B^2$. In fact, $2A^2 + 2B^2 - (A-B)^2 = (A+B)^2$ is positive semidefinite.
My question is: Is there a ...
- 123
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Does k(X) have a k-basis for every set X, without AC?
This question is inspired by Pace Nielsen's recent question Does a left basis imply a right basis, without AC?.
For any field $k$, the field $k(x)$ of rational functions in one variable has an ...
- 35.2k
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2
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The $r$-dimensional volume of the Minkowski sum of $n$ ($n\geq r$) line sets
Let $n$ line sets be $\mathcal{S}_i=\{a\mathbf{h}_i:0 \le a \le 1\}$, for $1 \le i \le n$, where $\{\mathbf{h}_1,\cdots,\mathbf{h}_n\}$ is a vector group of rank $r$ in the $r$-dimensional Euclidean ...
- 550
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2
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1k
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Stable conjugacy for integer matrices
$\DeclareMathOperator\GL{GL}$Let $F$ be a field, and $E$ an extension field. Then two matrices in $\GL_n(F)$ are conjugate if and only if they are conjugate in $\GL_n(E)$. I'm curious whether the ...
- 385
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Prove that matrix is positive definite
I faced a hard question in kernel methods theory, which I can't answer for about one week. Initially it was formulated in terms of positive valued functions, but it could be reformulated easier:
Let $...
user99354
12
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5
answers
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Is this formulation of the Singular Value Decomposition standard?
In customary formulations of the Singular Value Decomposition or SVD that I have seen,
(e.g., Wikipedia or Gil Strang's textbooks) it is always stated in terms of writing an
$m \times n$ matrix $M$ (...
- 15.3k
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votes
2
answers
9k
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What is the time complexity of the matrix exponential?
While trying to compute the Matrix Exponential of an $n \times n$ array I decided to take advantage of a Python function called scipy.linalg.expm().
According to ...
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Conjugacy in $GL(n,\mathbb Z)$
How can I determine whether $A_1,A_2\in GL(n,\mathbb Z)$ conjugate in $GL(n,\mathbb Z)$ and if they are, how can I find a $P\in GL(n,\mathbb Z)$ for which $A_2 = P^{-1}.A_1.P$ ?
In $GL(n,\mathbb Q)$ ...
- 347
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1
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Is there an eigenvalue of modulus larger than 1?
Given a matrix $A\in \operatorname{SL}_d(\mathbb{Z})$ (the special linear group) satisfying the two conditions: (1) no eigenvalue of $A$ is a root of unity, (2) the characteristic polynomial of $A$ is ...
- 43.2k
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Smallest set such that all arithmetic progression will always contain at least a number in a set
Let $S= \left\{ 1,2,3,...,100 \right\}$ be a set of positive integers from $1$ to $100$. Let $P$ be a subset of $S$ such that any arithmetic progression of length 10 consisting of numbers in $S$ will ...
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Representability of matroids over $\mathbb R$
Let $M$ be a matroid, for example viewed as being given by a finite set $X$ and a rank function $d : P(X) \to {\mathbb N}$ such that
1) $d(\varnothing)=0$, $d(\lbrace x \rbrace)=1$, for all $x \in X$,...
- 25.5k
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1k
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Eigenvalue perturbation theory via Feynman diagrams
Suppose I have a matrix given by a sum
$$A=D+\epsilon B$$
where $D$ is diagonal and $\epsilon$ is small, and I want the eigenvalues of $A$ as a power series in $\epsilon$. The first two orders in ...
- 1,549
12
votes
1
answer
902
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Positive 4-form
Denote by $W$ the space of all symmetric bilinear forms on $\mathbb{R}^n$.
Let $Q$ be a quadratic form on $W$.
Suppose that $Q(b)\geqslant 0$ for any $b\in W$ such that $b(X,Y)=\ell(X)\cdot\ell(Y)$ ...
- 45k
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elementwise functions of positive definite matrix
The fact that the Schur (that is, element wise) product of two positive definite (symmetric) matrices is positive definite immediately implies (using the convexity of the positive semi definite cone) ...
- 96.4k
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4
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Topologizing free abelian groups
For any set $S$ one can consider the free abelian group $\mathbb{Z}[S]$ generated by this set. Now suppose, there is a topology on $S$ given. Is it possible to find a topology on $\mathbb{Z}[S]$ in ...
- 11.1k
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What is known about the eigenvectors of the $2^n \times 2^n$ Hadamard matrix?
What is known about the eigenvectors of the $2^n \times 2^n$ Hadamard matrix defined recursively by $H_1=(1)$ and $$ H_N=\begin{pmatrix}H_{N/2} & H_{N/2} \\ H_{N/2} & -H_{N/2}\end{pmatrix}, $$ ...
- 1,324
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4
answers
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Real and quaternionic representations according to weights
According to this question, it is easy to know whether a (complex, finite-dimensional) representation is self-dual or not: just check if the weight distribution in space is symmetric about the origin.
...
- 2,091
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5
answers
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Infinite direct product of the integers not a free module over the integers [duplicate]
Possible Duplicate:
Is it true that, as Z-modules, the polynomial ring and the power series ring over integers are dual to each other?
Is there an easy proof? I only found citations but have no ...
12
votes
5
answers
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Analogue of Cayley Hamilton theorem for operators on Hilbert space
Is there an analogue of Cayley Hamilton theorem which holds for operators on a separable Hilbert space. Obviously the characteristic polynomial will be replaced by something else.
- 2,099
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1
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Closest 3D rotation matrix in the Frobenius norm sense
Given a 3 by 3 matrix $M$ I would like to find the rotation matrix $R$ minimizing the Frobenius norm:
\begin{equation}
\|R-M\|_F
\end{equation}
Is there a closed form solution for $R$, or is it ...
- 561
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2
answers
3k
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On the positive definiteness of a linear combination of matrices
In my work in PDE, the following problem in linear algebra came up. Any help in this direction is appreciated.
QUESTION:
Let $m,n\in\mathbb{N}$ and let $A_1,\ldots, A_m\in M_n(\mathbb{R})$ be real, ...
- 895
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votes
2
answers
970
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which norms can be realized as operator norms?
Assume $(V,∥∥_V),(W,∥∥_W)$ are both finite dimensional normed spaces. We have the induced operator norm on ${\rm Hom}(V,W)$.
It turns out that the operator norm is induced by an inner product iff ...
- 6,741
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votes
4
answers
752
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Additive commutators and trace over a PID
I would like to find an example of principal ideal domain $R$, such that there exists a square matrix $A\in \mathfrak{M}_n(R)$ with zero trace that is not a commutator (i.e. for all $B,C \in \mathfrak{...
- 2,829
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2
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2k
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Non-degenerate multilinear forms
Is there a standard notion of non-degeneracy for multilinear forms?
My motivation is simple curiosity, by the way!
- 47.3k