Linear algebra is concerned with vector spaces of all dimensions and linear transformations between them. This includes: systems of linear equations, Hamel basis, dimension, matrices, determinant, trace, eigenvalues and eigenvectors, diagonalization, etc. For questions specifically concerning ...
12
votes
0answers
291 views
The normalizer of $\mathrm{GL}(n,\mathbf Z)$ in $\mathrm{GL}(n,\mathbf Q)$
It seems that the normalizer of $H=\mathrm{GL}(n,\mathbf Z)$ in $G=\mathrm{GL}(n,\mathbf Q)$ is "almost" equal to itself, that is,
$$
N_G(\mathrm{GL}(n,\mathbf Z))=Z(G) \cdot \mathrm{GL}(n,\mathbf ...
10
votes
0answers
77 views
Combinatorics in finite vector space
Let $q$ be a prime power and $V$ a finite $\mathbb F_q$-vector space with two subspaces $I$ and $J$.
Let $k$, $a$ and $b$ be non-negative integers.
Determine the number of subspaces $K$ of $V$ ...
8
votes
0answers
152 views
Variety of pairs of product-zero matrices
Here's an old qualifying exam question I got stuck on. Consider the variety $X$ of pairs of matrices $(A,B)$ satisfying $AB = BA = 0$ (with entries in some field). What are the irreducible components ...
8
votes
0answers
191 views
Limit of sequence of growing matrices
Let
$$
H=\left(\begin{array}{cccc}
0 & 1/2 & 0 & 1/2 \\
1/2 & 0 & 1/2 & 0 \\
1/2 & 0 & 0 & 1/2\\
0 & 1/2 & 1/2 & 0
\end{array}\right),
$$
...
8
votes
0answers
177 views
Inverse of Toeplitz Matrix Property
Sorry if this question has been asked already but I didn't find it. Given a symmetric Toeplitz matrix of the form
$$\left[\begin{array}{llll}
a_0 & a_1 & \dots & a_n\\
a_1 & a_0 ...
8
votes
0answers
155 views
$\mathcal{O}(n,\mathbb R)$ spans $\mathcal{M}(n,\mathbb R)$
Let $n\geq 3$. One can show that the orthogonal group of degree $n$ over the real field, $\mathcal{O}(n,\mathbb R)$, spans the entire vector space of real $n\times n$ matrices, $\mathcal{M}(n,\mathbb ...
7
votes
0answers
887 views
Inverse of a Toeplitz Matrix
A Toeplitz matrix or diagonal-constant matrix is a matrix in which each descending diagonal from left to right is constant. For instance, the following matrix is an $n\times n$ Toeplitz matrix:
$$
A ...
6
votes
0answers
141 views
Rank of a $n! \times n$ matrix
Let $X=\langle x_1,\cdots,x_n \rangle$ and $Y=\langle y_1,\cdots,y_n \rangle$ be vectors of positive reals and let $P=\langle p_1,\cdots,p_n \rangle$ be a permutation of the numbers $\{1,\cdots,n\}$. ...
6
votes
0answers
270 views
Condition of an eigenvector problem
Please, somebody help me with this problem.
[Ciarlet 2.3-5] Let ${A}$ and ${B} = {A} + \delta{A}$ be two symmetric matrices with eigenvalues
$$\alpha_1\ \leq\ \alpha_2\ \leq\ \ldots\ \leq\ ...
6
votes
0answers
113 views
Prove the determinant of this matrix
We have a square matrix, that all elements on main diagonal are zero, and other elements are following:
$$a_{i,j}=\begin{cases}
1,&\text{if i+j belongs to Fibonacci numbers,}\\
0,&\text{if ...
6
votes
0answers
138 views
Alternating two-form
Let $V$ be a real vector space of dimension $2$, and let $\langle\ \ ,\ \ \rangle$ be an inner product on $V$. Define $f:V^4 \to \mathbb{R}$ by $$f(x,y,z,w):=\langle x,y \rangle \langle z,w \rangle- ...
6
votes
0answers
241 views
Generalized Eigenvalue Problem with one matrix having low rank
I have a specific Generalized Eigenvalue Problem (GEVP) where i am primary not interested in solving this problem but concluding from a standard EVP the spectrum of the GEVP.
The Problem
Let $A$ be ...
6
votes
0answers
85 views
Piecewise Affine Bijections of $\mathbb{R}^n$
I have a min-max function $f:\mathbb{R}^n\to\mathbb{R}^n$ of the form $$f(x) = \min_{i=1,\dots,n}\max_{j=1,\dots,n}(\alpha_{ij}^Tx + \beta_{ij})\quad\text{where each } \alpha_{ij}\in ...
6
votes
0answers
175 views
Uses of Chevalley-Warning
In the recent IMC 2011, the last problem of the 1st day (no. 5, the hardest of that day) was as follows:
We have $4n-1$ vectors in $F_2^{2n-1}$: $\{v_i\}_{i=1}^{4n-1}$. The problem asks :
Prove the ...
5
votes
0answers
40 views
Prove that if $Q^tQ = I$ and $A = QR$, then $\|Ax - b\| = \|Rx - Q^tb\|$
I have a linear algebra final tomorrow and was practicing a few proofs. I want to make sure this proof is correct.
Prove that: If $Q^tQ = I$ and $A = QR$, then $\|Ax - b\| = \|Rx - Q^tb\|$
...
5
votes
0answers
84 views
How can I tell if a matrix can be LU decomposed without actually finding the L, U?
I've seen quite a few problems like that.
For example, suppose we have the following A matrix:
\begin{pmatrix}
5 & 1 & 1 & 1 & 0 &1\\
2 & 6 & -1 & 0 & -1 ...
5
votes
0answers
127 views
Monotone matrix norms
[Ciarlet 2.2-10]
Let $\mathscr{S}_n$ be the set of symmetric matrices and $\mathscr{S}_n^+$ the subset of non-negative definite symmetric matrices. A matrix norm $\|\cdot\|$ to be monotone if
...
5
votes
0answers
55 views
My proof that if for a k degree polynomial $P(x)$, for the matrix $A$, $P(A)=0$ then $A$ is invertible
Let $P(x)$ be a $k$-degree polynomial with with non-zero free coefficient. Prove that if for matrix $A$, $P(A)$=0, then $A$ is invertible and $A^{-1}$ is $k-1$ degree $A$ polynomial.
Here's my ...
5
votes
0answers
69 views
Cauchy-Schwarz inequality for bilinear forms valued in an abstract vector space
This question is perhaps a little vague; part of what I want to know is what question I should ask.
First, recall the following form of the Cauchy-Schwarz inequality: let $V$ be a real vector space, ...
5
votes
0answers
51 views
Volume of the intersection of ellipsoids
How do I compute the volume of the intersection of two n-dimensional ellipsoids?
Given an $n$-vector $c$ and a symmetric positive-definite $n\times n$ matrix $A$, define the ellipsoid ...
5
votes
0answers
87 views
How is $\mathrm{PGL}(V)$ a subgroup of $\mathrm{P\Gamma L}(V)$?
I've stumbled upon a strange exercise while reading "Notes on Infinite Permutation Groups" by Bhattacharjee, Möller, Macpherson and Neumann. If you have the book, the exercise is 7(ix) on page 66.
...
5
votes
0answers
78 views
numerical linear algebra tricks for repeated sums and inversions with symmetric positive-definite matrices
I'm doing the following procedure to get the max-likelihood estimate of a matrix-variate normal distribution from $r$ samples of matrices in $\mathbb{R}^{n \times p}$ (algorithm from Dutilleul ...
5
votes
0answers
92 views
Linear algebra estimates
Here is something that has been troubleing me lately. I don't know if it is true of not. I suspect it is.
Suppose that $A,B$ are two $n \times n$ matrices with complex entries. $A^t = A$, $\bar B^t = ...
5
votes
0answers
184 views
Approximating commuting matrices by commuting diagonalizable matrices
Suppose the matrices $A$ and $B$ commute. Do there exists sequences $A_n$ and $B_n$ of matrices such that
$A_n \rightarrow A$, $B_n \rightarrow B$.
Each $A_n$ is diagonalizable and the same for ...
5
votes
0answers
96 views
Reference suggestion: eigenvalues of tridiagonal matrices
I would like to ask for a reference on the problem of computing the eigenvalues/eigenvectors of tridiagonal matrices (not necessarily with constant diagonals).
I have seen authors use continued ...
5
votes
0answers
71 views
Continuous choice of basis for subspaces
Consider the flag variety (or flag manifold, depending on who you are) $V=\mathrm {Fl} (3,\mathbb C)$ of complete flags of subspaces of $\mathbb C^3$. That is, an element of M is a tuple (L , P) ...
5
votes
0answers
116 views
Symmetric functions of the eigenvalues of A+B, A, B, ABA, BAB, et.c.
(this is an improved version of What about other symmetric functions of the eigenvalues? )
Let $A$ be a matrix with eigenvalues $\lambda_1, \dots, \lambda_n$. Then $\det(A) = \lambda_1 \dots ...
5
votes
0answers
147 views
Points and lines covering them
Let $n$ be a positive integer. A subset $S$ of points in plane satisfies the following conditions:
a) We can't find $n$ lines in plane, such that every element of $S$ belongs to at least one of these ...
5
votes
0answers
227 views
Pfaffian properties
Given a $2n\times 2n$ real skew-symmetric matrix $A$, its Pfaffian $\mathrm{Pf}$ is defined as:
$$
\mathrm{Pf}(A) = \frac1{2^n n!}\sum_{\sigma\in S_{2n}}\mathrm{sgn}(\sigma)\prod_{i=1}^n ...
5
votes
0answers
141 views
Solutions to equation in matrix form
Suppose $x=[x_1, x_2, \cdots, x_n]^t$, $b=[b_1,b_2,\cdots,b_n]^t$ with $b_i\in K$ and $A\in M_n(K)$, where $K$ is a field. There are well known criteria for the system of equations $Ax=b$, by ...
5
votes
0answers
142 views
linear system solution, iterative vs direct
Dear all,
I have systems like
$(A - \lambda B) X = F$
where lambda is being updated inside a loop. I also have a limited number of eigenvectors of the matrix pair (A, B), say 40 eigenpair from a ...
5
votes
0answers
150 views
A relation between permanents and determinants
I have skimmed this video that I found on mathoverflow:
http://tube.sfu-kras.ru/video/407?playlist=397
At about 15:05 the lecturer wrote down an equality
$\sum F(m_1, \ldots, m_m)z^{m_1}\ldots ...
4
votes
0answers
54 views
“Convex” polynomials
Let me define "convex" polynomials, as the smallest class $\mathcal{C}$ of functions $p:\mathbb{R}\rightarrow \mathbb{R}$ defined (inductively) as:
UPDATED (case 0 was missing):
0) $p(x)=x$, i.e., ...
4
votes
0answers
48 views
Invariant of matrix under elementary transformations
$\DeclareMathOperator{\rank}{rank}$
Let $A \in \mathbb R^{n \times n}$, $b \in \mathbb R^n$, $c \in \mathbb R$. Consider the following matrix
$$
B = \begin{bmatrix} A & b \\ b^T & c ...
4
votes
0answers
25 views
Isomorphism between $E_8$ lattice and lattice defined by Extended Hamming Code
I have read that the following two lattices are isomorphic, and of course it seems believable, but it would be nice to have a sketch of how to construct the bijection.
Let $C$ be some extended ...
4
votes
0answers
47 views
When are two commuting linear operators functions of each other
I've computed that the following is valid for certain functions but I've hit a slight bump in my proof. I was wondering if someone could clear it up.
If we formally consider the integral operator $E ...
4
votes
0answers
66 views
Probability binary Toeplitz matrix invertible
A Toeplitz matrix or diagonal-constant matrix is a matrix in which each descending diagonal from left to right is constant.
What is the probability that a random $n \times n$ binary Toeplitz ...
4
votes
0answers
57 views
Linear Independence Game
Suppose you have a set $X$ of vectors in $\mathbb{F}_2^n$, with $|X| \ge n+1$, and consider the following game. On their turn, each player (2 player game) chooses from $X$ one vector and sets it aside ...
4
votes
0answers
74 views
A basis of the symmetric power consisting of powers
Let $V$ be a complex vector space of dimension $n$. Denote by $v_1\odot\cdots\odot v_k$ the image of $v_1\otimes\cdots\otimes v_k$ in the symmetric power $\newcommand{\Sym}{\mathrm{Sym}}\Sym^k(V)$. It ...
4
votes
0answers
44 views
In stating that the union of vector subspaces is a subspace iff they are ordered, why require $F$ finite?
On the bottom of page 38 of Roman's Advanced Linear Algebra is written the following (here $V$ is a vector space over the field $F$ and $\mathcal{S}(V)$ is the set of linear subspaces of $V$):
"...if ...
4
votes
0answers
82 views
Geometric intuition for Jordan normal forms (invariant subspaces, shearing, scaling, etc.)
I'm trying to visualize what a linear operator does to a vector space if that operator can be put into Jordan normal form.
For concrete motivation, let's take $V = \mathbb{R}^3$, with some linear ...
4
votes
0answers
106 views
Solving a system of linear inequalities
I have a personal problem I want to solve. I have a system of linear inequalities with 97 unknowns and 150000 ineqaulities, I think formal notation of the problem should be something like this
...
4
votes
0answers
153 views
Understanding a proof about Hilbert Matrix
EDIT: I asked 3 questions. The first one I was able to solve myself, and the other two I cross-posted to MO.
Lately I've been interested in the Hilbert Matrix (its definition will come later). I went ...
4
votes
0answers
122 views
On the integer feasibility of polytopes defined by idempotent integer matrices
EDIT: I realized that while writing this question, I was reasoning about orthogonal projections. Thus, I forgot to transpose when forming the projection on to the space orthogonal to the image of $P$. ...
4
votes
0answers
183 views
Recovering a Matrix knowing its eigenvectors and eigenvalues
Given the eigenvalues and eigenvectors of a matrix $R^{n\times n}$ is that possible to recover the same matrix from smaller matrices $R^{(n-1) \times (n-1)}$ where one of its eigenvalues and ...
4
votes
0answers
140 views
Computing the SVD factorization on C++ (using the proof of the existence of the SVD factorization)
I am doing a C++ program that computes the SVD factorization of a real matrix A without using any known library of algebra that contains the implementation. In addition, QR descomposition is not ...
4
votes
0answers
67 views
Condition number of $A^{-1}B$ where $A$ and $B$ are banded toeplitz matrices.
I'm looking at a filtering problem with feedback, which can be represented by the equation $A\underline{y} = B\underline{x}$, where $A$ and $B$ are lower triangular banded toeplitz matrices and ...
4
votes
0answers
63 views
Jordan decomposition of the action of Jordan blocks on alternating tensors
Let $J := J_r(\lambda)$ be an $r \times r$ Jordan block with complex eigenvalue $\lambda \neq 0$, and consider it acting on $V := \mathbb{C}^{r}$ in the usual way. Then we have the induced map:
...
4
votes
0answers
120 views
Proof that the Arf invariant is independent of choice of basis
I'm confused about the proof of the following claim:
Let $V$ be a vector space of dimension $2n$ and let $e_i, f_i$ be a symplectic basis. Let $q: V \to Z_2$ be a non-degenerate quadratic form. ...
4
votes
0answers
209 views
Is the “Constant Rank Theorem” the same as the “Domain Straightening Theorem”? Which theorem is which?
Wikipedia says that the inverse function theorem is a special case of the "constant rank theorem".
I'm pretty sure this is supposed to be the same theorem as the "Rank Theorem" on p. 47 of Boothby ...
