Questions tagged [linear-algebra]
Questions about the properties of vector spaces and linear transformations, including linear systems in general.
5,661
questions
11
votes
2
answers
551
views
Classification of algebras of finite global dimension via determinants of certain 0-1-matrices
I restrict to the elementary problem that is equivalent to give a classification when Morita-Nakayama algebras have finite global dimension (see the end of this post for some background).
A Morita-...
2
votes
2
answers
257
views
Equal-valued determinants in search of a proof: Part III
Encouraged by David's proof for my earlier MO question, let's consider a similar problem.
I can prove the below equality by computing each of the two sides, directly. That means, there is an ...
11
votes
1
answer
569
views
Catalan determinants in search of a proof: Part II
This problem involves the Catalan numbers $C_n=\frac1{n+1}\binom{2n}n$.
I can prove the below equality by computing each of the two sides, directly. That means, there is an algebraic proof.
...
7
votes
1
answer
688
views
Is Gram-Schmidt on a separable Hilbert space operator norm continuous?
Let $\mathcal H$ be a separable Hilbert space, with inner product $\langle\cdot,\cdot\rangle$, and with orthonormal basis $(e_i)_{i\in\mathbb N}$. Consider a continuous linear embedding $A\colon\...
5
votes
1
answer
407
views
Determining the primitive order of a binary matrix
Let ${\bf A}_n$ be an $2n \times 2n$ matrix that is defined as follows
$$
{\bf A}_n=\left(
\begin{array}{c}
0&0&\cdots&0&0&0&0&1&1\\
0&0&\cdots&0&0&...
5
votes
1
answer
1k
views
eigenvalues of a symmetric matrix
I have a special $N\times N$ matrix with the following form. It is symmetric and zero row (and column) sums.
$$K=\begin{bmatrix}
k_{11} & -1 & \frac{-1}{2} & \frac{-1}{3} & \frac{-1}{4}...
7
votes
1
answer
496
views
Cycle types of permutations from affine group
Let $V$ be a vector space of dimension $n$ over the field $F=\mathrm{GF}(2)$. We identify $V$ with the set of columns of length $n$ over $F$. Let $G = \mathrm{AGL}(V)$ be a group of affine ...
3
votes
1
answer
76
views
Dimension of fixed vectors of a semi-linear operator
Let $L$ be a field with a field embedding $\sigma:L \rightarrow L$, and $K=L^{\sigma}$ be the fixed field of $\sigma$. For $A \in M_n(L)$ a matrix, consider the set $X=\{x \in L^n|Ax=\sigma(x) \}$ ...
4
votes
0
answers
380
views
Trace of the adjoint action is an eigenvalue in $\mathrm{U}(L)$?
Let $L$ be a finite-dimentional complex Lie algebra. $\forall x \in L$, one defines the adjoint action of $x$ on $L$ as the map
$\mathrm{ad}_x : L \to L, \text{ with } \mathrm{ad}_x(y) = [x,y]$
for ...
8
votes
1
answer
480
views
What is the minimum $k$ such that $A^k \equiv I$ mod p for invertible matrices?
Let $F$ be a finite field of order $p$, where $p$ is prime. For any $n\times n$ matrix $A$ that is invertible over $F$, then there would appear to exist integers $k$ such that $A^{k} = I$. My question ...
0
votes
1
answer
242
views
Perturbing a normal matrix
Let $N$ be a normal matrix.
Now I consider a perturbation of the matrix by another matrix $A.$
The perturbed matrix shall be called $M=N+A.$
Now assume there is a normalized vector $u$ such that $\...
5
votes
1
answer
218
views
Stable matrices and their spectra
I am a graduate student in engineering and we work a lot with so-called Hurwitz (or stable) matrices.
A matrix in our terminology is called stable if the real part of the eigenvalues is strictly ...
2
votes
1
answer
186
views
Question on the proof of any finite dimensional module of a semisimple Lie algebra is semisimple
I'm going through a proof of the theorem, any finite dimensional module of a semisimple Lie algebra is semisimple, from page 10 of these notes (pdf). I am having a difficult time understanding few ...
6
votes
1
answer
416
views
If a commutative graded algebra is free over a graded subalgebra, then must it have a graded basis?
Fix a field $\mathbf{k}$ and an $\mathbb{N}$-graded commutative $\mathbf{k}$-algebra $A = \bigoplus\limits_{n = 0}^{\infty} A_n$ of finite type. ("Finite type" means that each $A_n$ is a ...
4
votes
1
answer
280
views
Which groups can be reconstructed from a single invariant subspace?
Let $G\subseteq\mathrm{Perm}(\Bbb R^n)$ be a matrix group consisting of permutation matrices acting on $\Bbb R^n$. Let $U\subseteq\Bbb R^n$ be an irreducible invariant subspace w.r.t. $G$. Now, define ...
1
vote
0
answers
54
views
When does the multi-spectral radius coincide with the spectral radius of the sum of linear transformations?
Suppose that $X$ is a finite dimensional Hilbert space and
$A_{1},\dots,A_{r}:X\rightarrow X$ are linear transformations. Define the multi-spectral radius $\rho(A_{1},\dots,A_{r})$ to be
$$\limsup_{n\...
2
votes
1
answer
485
views
An inequality on elementary symmetric polynomial of eigenvalues
For $A \in \mathbb{R}^{n\times n}$, I am wondering if the following inequality holds,
$$\text{tr}(A)^2 - \text{tr}(A^2) \leq ||A||_*^2 - ||A||_F^2$$
This is equivalent to the following inequality on ...
3
votes
1
answer
764
views
Real part of eigenvalues and Laplacian
I am working on imaging and I am a bit puzzled by the behaviour of this matrix:
$$A:=\left(
\begin{array}{cccccc}
1 & 0 & 0 & -1 & 0 & 0 \\
0 & 0 & 0 & 0 & -1 &...
3
votes
1
answer
421
views
Spectrum of this block matrix
Consider the following block matrix
$$A = \left(\begin{matrix} B & T\\ T & 0 \end{matrix} \right)$$
where all submatrices are square and
matrix $B = \mbox{diag}\left(b_1 ,0,0,\dots,0,b_n \...
0
votes
1
answer
115
views
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]...
0
votes
0
answers
162
views
Smallest eigenvalues of block Kronecker product
Let $D \in \mathbb{R}^{n \times n}$ defined as
\begin{equation}
D := \begin{pmatrix}
1 & 0 & \cdots & \cdots & 0 \\
-1 & 1 & \ddots & \ddots & 0 \\
\vdots & \ddots &...
3
votes
0
answers
151
views
Reference request: invariants/tableaux functions for 4 lines in $P^3$
Does anybody have a reference for invariants of configurations of linear subspaces in the projective space?
In particular I would be curious to see an explicit expression of the invariant functions ...
5
votes
1
answer
202
views
Spectral radius for multiple linear operators
Suppose that $X$ is a finite dimensional Hilbert space. Let $A_{1},\dots,A_{r}:X\rightarrow X$ be linear operators. Then define the multi-spectral radius of $(A_{1},\dots,A_{r})$ to be
$$\limsup_{n\...
0
votes
0
answers
227
views
What matrix has only negative or zero real part for all the eigenvalues?
Say $X \in \mathbb{R}^{m\times m}$,
Is it possible to have a constraint on $X$, such that all the eigenvalues has negative or zero real part?
What I conjecture
The following $X$ has only negative ...
3
votes
2
answers
1k
views
What do the eigenvectors of the $n$th roots of $I_n$ look like?
This was asked at math stackexchange a long time ago with no answers but some upvotes.
Let $A^n=I,$ where $A$ is $n\times n,$ and assume that $A^k\neq I,$ for all $1\leq k<n.$ Since its ...
3
votes
0
answers
254
views
Equal principal minors of matrix plus rank-1 and inverse
Given an invertible real matrix $A$ and real column vectors $b$ and $c$.
For which $A$, $b$ and $c$ are all corresponding principal minors of $B = A-bc^T$ and
$A^{-1}$ equal?
According to a ...
0
votes
0
answers
35
views
What is the locus defined by those equations?
I would like to know what is the locus of $x \in \Bbb R_+^n$ ($n=2$ would already be fine) defined by
$\sum a_i \cdot x_i$ s.t. $a_i+\epsilon \geq 0$, $\epsilon \in \Bbb R$.
I know that if $\...
4
votes
2
answers
349
views
Linear operator on polynomials and invariant sets of roots
Let $T:\mathbb{C}_n[x] \to \mathbb{C}_n[x]$
be a linear map from the vector space of polynomials of degree $n$ to itself.
Let $S \subset \mathbb{C}$ be a set with at least $3$ points, such that
for ...
3
votes
1
answer
2k
views
When is the matrix norm multiplicative
Let $|| = ||_{p,q}$ be an operator norm on $\mathbb R^{n \times m}$. In General, $\|AB\|\le \|A\|\|B\|$. Is there some criterion on $A, B$ (at least for some operator norms) so that $\|AB\| = \lVert ...
3
votes
1
answer
59
views
Looking for an Alternative Characterization of the Upper Orthant of a Convex Hull
I have a question on convex analysis, which I require in one step of my research work. Before stating it, let me give a small background.
Let $Y, X_1,\ldots,X_n$ be $n+1$ points in $\mathbb{R}^d$. ...
3
votes
2
answers
1k
views
Completely positive matrix with positive eigenvalue
A matrix $A \in \mathbb{R}^{n \times n}$ is called completely positive if there exists an entrywise nonnegative matrix $B \in \mathbb{R}^{n \times r}$ such that $A = BB^{T}$.
All eigenvalues of $A$ ...
0
votes
1
answer
43
views
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$$
where ${\bf x}_n \in \mathbb{R}^m (\forall n)$, and ${\bf A} \in \mathbb{R}^{m\times m}$ and ${\...
10
votes
2
answers
979
views
Determinantal symmetry: proof requested: Part I
Consider the determinantal function
$$F(a,b,c):=\det\left[\binom{i+j+a+b}{i+a}\right]_{i,j=0}^{c-1}.$$
I would like to ask:
QUESTION. Can you provide an argument, combinatorial or otherwise, to ...
5
votes
9
answers
7k
views
Applications of basic linear algebra concepts to computer science? [closed]
I'm trying to explain linear algebra to some programmers with computer science backgrounds. They took a course on it long ago, but don't seem to remember much. They can follow basic formalism, but ...
8
votes
1
answer
472
views
A question about special linear group
Is there any way to find all matrices $G \in SL(n,\mathbb Z)$ such that there exists a matrix $A \in GL(n,\mathbb R)$ satisfying
$$
AGA^{-1} \in SO(n,\mathbb R)?
$$
1
vote
0
answers
154
views
Properties of vector combinations in the non-negative orthant
Given a vector $x \in \mathbb{R}^{n}_{0+}$ such that $x = \sum^{k}_{i=1} \alpha_{i}v_{i}$, the vectors $(v_{1},...,v_{k}) \in \mathbb{R}^{n}_{0+}$ are an independent set, $k < n$, and $\alpha_{i} &...
2
votes
0
answers
183
views
Deriving category of quadratic functors
Recall that a fuctor $T: \cal C \to \cal A$ from pointed small $\cal C$ with coproducts to additive and Karoubian — or, even better, abelian $\cal A$ is called quadratic if kernel of sum of obvious ...
8
votes
1
answer
170
views
Distance between subalgebras and positive elements in matrices
I repost here from stackexchange, as I was not given an answer there. (https://math.stackexchange.com/questions/2956530/distance-between-subalgebras-and-commutants-in-matrix-algebras)
This is a ...
6
votes
0
answers
367
views
Is there an easy way to compute the maximum isotropic subspace over finite fields?
Given a quadratic form (or a symmetric $n \times n$ matrix $A$), an isotropic subspace is a subspace $U$ such that $$U^t A U=0,$$
If I am not mistaken, when the matrix is over reals, the maximum ...
2
votes
0
answers
224
views
Quartic optimization problem over the unit Euclidean sphere
I want to solve following optimization problem in $x \in \mathbb R^n$.
$$\begin{array}{ll} \text{maximize} & \displaystyle\sum_i(x M_i x^T)^2\\ \text{subject to} & \|x\|_2 = 1\end{array}$$
...
1
vote
1
answer
125
views
Expected norm of linear maps
I want to compute the expected norm of a vector-matrix multiplication. I have a vector $x \in \mathbb{R}^n$ with norm one and a matrix $M \in \mathbb{R}^{n \times n}$, whose entries are iid taken from ...
0
votes
1
answer
142
views
Pre-garbling does not improve capacity of a channel
Suppose ABC=B for some column stochastic matrices A, B, and C.
Can the following implication be made without further restrictions:
There necessarily exists a column stochastic matrix D such that DB=BC?...
4
votes
2
answers
156
views
$\sum_{k=1}^dA_k^*A_k$ and $\sum_{k=1}^dA_kA_k^*$ have the same norms if $A_k$ are commuting
Let $E$ be a complex Hilbert space and $\mathcal{L}(E)$ be the algebra of all operators on $E$.
Let $A_1,\cdots,A_d$ be pairwise commuting operators on $E$. Is the equality
$$\left\|\displaystyle\...
2
votes
1
answer
435
views
Coordinate free expression for the determinant of a $2 \times 2$ matrix
Let $E\rightarrow X$ be a rank two vector bundle on a variety $X$. How can one write an abstract map $Sym^2(E)\rightarrow (\wedge^2 E)^{\otimes 2}$ that in local coordinates, when we fix a frame of $E$...
-2
votes
1
answer
152
views
About intersections of two totally isotropic subspaces fo a quadratic form [closed]
Let $Q$ be a quadratic form on $\mathbb R^{2m}$ with the signature $(m,m)$. The maximal totally isotropic subspaces in $(\mathbb R^{2m},Q)$ have then dimensions $m$.
What dimensions $1,...,m-1$ of ...
2
votes
2
answers
111
views
Correlation between the first and a random position of an ergodic bit sequence
Edit: Since the geometric approach did not work, I try now another approach: phrasing the problem as a quadratic programme.
Probabilistic version.
Let $x=(x_1,x_2, \ldots) $ be an ergodic random ...
5
votes
1
answer
93
views
On a maximum of a determinant with dependent variables
Let $x_1,\ldots,x_n\in [-1,1]^n$ and define the function
$$f(x_1,\ldots,x_n):= \prod_{i=1}^n\prod_{j=i}^n\left(1-\prod_{k=i}^j x_k\right).$$
This is a positive function, and actually coincides with ...
1
vote
1
answer
178
views
Subbundle generated by linearly dependent sections
On $\mathbb{P}^1$ consider the trivial bundle $\mathcal{O}\oplus \mathcal{O}$, and the subbundle $\mathcal{L}_{a,b}\subset\mathcal{O}\oplus \mathcal{O}$ that on an open subset $U$ of $\mathbb{P}^1$ is ...
3
votes
1
answer
3k
views
Why is the matrix of all 1's called "J"? [closed]
I've seen J referenced recently in some discussion about algebraic combinatorics, and it took me a while to figure out it was the matrix of all ones. It came up without definition, and I spent too ...
2
votes
0
answers
70
views
An two-norm estimate for symmetric $k$-tensors
Let $(V, \langle, \rangle)$ be an $n$ dimensional innerproduct space and let $S^k(V)$ denote the space of $k$-fold symmetric tensors. The inner product naturally extends to $S^k(V)$. Denote the ...