For any topic related to matrices. This includes: systems of linear equations, eigenvalues and eigenvectors (diagonalization, triangularization), determinant, trace, characteristic polynomial, adjugate, transpose, Jordan normal form, matrix algorithms (e.g. LU, Gauss elimination, SVD, QR), ...
8
votes
0answers
192 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
$$\left[\begin{array}{llll}
a_0 & a_1 & \dots & a_n\\
a_1 & a_0 ...
8
votes
0answers
157 views
$\mathcal{O}(n,\mathbb R)$ spans $\mathcal{M}(n,\mathbb R)$
Let $n\geq 3$. One can show that the orthogonal group of degree $n$ over the real field, $\mathcal{O}(n,\mathbb R)$, spans the entire vector space of real $n\times n$ matrices, $\mathcal{M}(n,\mathbb ...
7
votes
0answers
119 views
Fundamental domain of $\operatorname{GL}(n,\mathbb R)$ acted on by $\operatorname{GL}(n, \mathbb Z)$
What is a simple description of a fundamental domain of $\operatorname{GL}(n,\mathbb R)$ acted on by $\operatorname{GL}(n,\mathbb Z)$?
$\operatorname{GL}(n,\mathbb R)$ is the group of all real ...
6
votes
0answers
271 views
Condition of an eigenvector problem
Please, somebody help me with this problem.
[Ciarlet 2.3-5] Let ${A}$ and ${B} = {A} + \delta{A}$ be two symmetric matrices with eigenvalues
$$\alpha_1\ \leq\ \alpha_2\ \leq\ \ldots\ \leq\ ...
6
votes
0answers
111 views
Determining sign(det(A)) for nearly-singular matrix A
Motivation: determining whether a point $p$ is above or below a plane $\pi$, which is defined by $d$ points, in a $d$-dimensional space, is equivalent to computing the sign of a determinant of a ...
6
votes
0answers
175 views
Representation for Vandermonde's permanent
Permanent of a matrix A = $\|a_{i,j}\|_{i,j=1}^{n}$ is defined as
$$
\mathrm{Perm}(A) = \sum\limits_{\sigma \in S_{n}} a_{1,\sigma_{1}},\ldots,a_{n,\sigma_{n}}
$$
Is there some representation for ...
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
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
39 views
How many Matrices exist with increasing row and increasing column condition?
Given $N$, I would like to know the number of matrix constructed from $1$ to $N$ which satisfies the following condition:
1. The each row entries should be in increasing order.
2. The each column ...
5
votes
0answers
114 views
Proof that the set of doubly-stochastic matrices forms a convex polytope?
Does the set of all doubly-stochastic matrices form a convex polytope? In general, I wonder how the proofs of convexity and geometry can be established for sets of matrices of this kind? Anything to ...
5
votes
0answers
93 views
counterexample for: integral forms that are equivalent as rational forms are also equivalent as integral forms
I have the feeling that I'm missing something very obvious: I'm looking for a counterexmple for the following statement for some $n>1$ (it is trivially true for $n=1$):
Let $A,B\in\mathbb ...
5
votes
0answers
84 views
Uniqueness of an infinite system of linear ODEs
How to prove that $\dot{x}=ax,\space x(0)=1$ has a unique solution if $a,x$ are infinite dimensional matrices?
More specifically, let $Q$ be a bounded infinitesimal generator, i.e. ...
5
votes
0answers
227 views
Pfaffian properties
Given a $2n\times 2n$ real skew-symmetric matrix $A$, its Pfaffian $\mathrm{Pf}$ is defined as:
$$
\mathrm{Pf}(A) = \frac1{2^n n!}\sum_{\sigma\in S_{2n}}\mathrm{sgn}(\sigma)\prod_{i=1}^n ...
5
votes
0answers
388 views
Is there general formula for the exponential of a tridiagonal matrix?
For an arbitrary tridiagonal matrix of the form
$$ A = \begin{pmatrix} b_1 & c_1 & 0 & 0 & ... \\ a_2 & b_2 & c_2 & 0 & ... \\ 0 & a_3 & b_3 & c_3 & ...
4
votes
0answers
30 views
Gelfand's formula, different field
Gelfand's formula says that for a complex matrix $A \in \mathbb{C}^{n \times n}$, $$\rho(A) = \lim_{m \rightarrow \infty} \|A^m\|^{1/m},$$ where $\rho$ is the spectral radius (norm of maximal ...
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
$$
B = \begin{bmatrix} A & b \\ b^T & c ...
4
votes
0answers
66 views
Probability binary Toeplitz matrix invertible
A Toeplitz matrix or diagonal-constant matrix is a matrix in which each descending diagonal from left to right is constant.
What is the probability that a random $n \times n$ binary Toeplitz ...
4
votes
0answers
39 views
Classification of $n \times n$ matrix in which each two components differ in each row and column up to automorphism
Suppose, we define a class $A$ of $n \times n$ matrix as follows: $$\text{In each Row }i, \text{for any} j,k\ (1 \leq j,k \leq n )\ a_{ij} \neq a_{ik} $$ $$\text{In each Column }l, \text{for any } ...
4
votes
0answers
97 views
estimation of transition probabilities from aggregate data
Please, O mathematicians, help me understand the approach to the problem of estimating transition probabilities given only aggregate data in Kalbfleisch & Lawless' 1984 paper "Least-Squares ...
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
244 views
How to solve matrix equation $AX+XB=C$ for $X$
How to solve matrix equation $AX+XB=C$ for $X$?
It seems not too difficult, I tried many times but fail.
I'm an adult student... I am now vexed about Gilbert Strang - An Introduction to Linear ...
4
votes
0answers
59 views
Is there a way to exploit the fact that the covariance matrix has a blocked structure to more easily compute the multivariate normal density?
I'm trying to minimize the (negative) multivariate normal log likelihood (dropping constants):
$$
\log |\boldsymbol\Sigma|\,+(\mathbf{x}-\boldsymbol\mu)^{\rm ...
4
votes
0answers
133 views
What is the exponential generating function of the inverse matrix of an integer triangle?
Let ${Z}$ denote an integer triangle (like Pascal's), ${Z}_{n,k}$ for $0\leq k \leq n$ and let $f$ be an exponential generating function for polynomials $p_{n}(x)$ with $[x^k]p_{n}(x)= Z_{n,k}$.
...
4
votes
0answers
69 views
A MatrixExp question: simplifying $\int_0^t e^{A(t-t')} e^{A^T (t-t')} dt'$ for a real matrix A
I am interested in computing the following integral of a matrix exponential.
\begin{equation}
\int_0^t e^{A(t-t')} e^{A^T (t-t')} dt'
\end{equation}
The only assumption is that $A_{n\times n}$ is ...
4
votes
0answers
114 views
Product of matrices; MAPLE giving a strange answer
Either my brain is seriously fried up right now or the computer is wrong.
If I have a matrix $\begin{bmatrix}
4 & -2\\
2 & -1 \\
0 & 0
\end{bmatrix}$ multiply by its transpose ...
4
votes
0answers
217 views
Fourier matrix - multiplicity of eigenvalues?
This question is Miscellaneous Exercise M.10 in Chapter 8 (Bilinear Forms) of Artin's Algebra. (The sentences in italics are due to me.)
The row and column indices in the $n \times n$ Fourier ...
4
votes
0answers
151 views
Finding conjugacy classes of $PGL_{2}(\mathbb{F}_{q})$
Assume $q$ is odd. How does one go about finding the conjugacy classes of $PGL_{2}(\mathbb{F}_{q})$? I know that for $GL_{2}(\mathbb{F}_{q})$, one can consider the possible Jordan Normal Forms of the ...
4
votes
0answers
271 views
An optimization problem involving orthogonal matrices
Let $X\in\mathbb{R}^{3\times 3}$ be an orthogonal matrix. Then $\mathrm{vec}X\in\mathbb{R}^9$ is a 9 by 1 vector formed by stacking the columns of the matrix $X$ on top of one another. Given a matrix ...
4
votes
0answers
208 views
Evaluating matrix-continued fractions?
I have a matrix-valued continued fraction defined in the following way: $\alpha_n$ and $\beta_n$ are matrices, and I am interested in the quantity
$A_1$, where all the $A_n$, $n = 1, 2, \dots$ are ...
4
votes
0answers
90 views
image of symmetric matrices under representation of $GL_2(\mathbb{R})$
Let $W$ be a real vector space of dimension $2$ and let $\rho_k:GL_2(\mathbb{R}) \to GL(\mathbf{S}^kW)$ be the standard representation of $GL_2(\mathbb{R})$. Since $\rho_k$ is polynomial, it naturally ...
3
votes
0answers
28 views
Definition and some elementary properties of the “vector turn map”
This is actually a follow up question. I have to apologise for the length of it. I didn't anticipate to be that long. I hope that it will be proved interesting nevertheless. I used the name "vector ...
3
votes
0answers
27 views
If the matrix is positive definite, then its similar matrix is also positive definite?
If $A$ is positive definite and $B$ is similar to $A$.
Can we say that $B$ is also positive definite?
I guess it is true since two matrices have same eigenvalues, and if $\sigma(A) > 0$, and so is ...
3
votes
0answers
48 views
Name of a special matrix
I have a matrix which is kind of symmetrical with the other diagonal, i.e., something like
$$A = \left[ \begin{array}{c c c c}
a & b & c & d \\
e & f & g & c \\
h ...
3
votes
0answers
59 views
How to check whether it is possible to solve this problem?
We have a matrix with numbers. We can add $1$ to any selected element and this action adds $1$ to elements according to some function, which I'll call the $X$-function. For instance, $1$ could be ...
3
votes
0answers
43 views
How this martix the eigenvalue and Jordan canonical form
let $a_{i,j}\in R,A=(a_{i,j})_{n\times n} $,and
$a_{i,j}=\begin{cases}
1&i+j\in\{n,n+1\}\\
0&i+j\notin\{n,n+1\}
\end{cases}$
that's meaning:
$$A=\begin{bmatrix}
...
3
votes
0answers
70 views
How many discrete subgroups does the Heisenberg group have?
Is there an easy way to describe an arbitrary discrete group in the Heisenberg group?
I figured that at least the family
$$
\begin{pmatrix}
1 & x\mathbb Z & z\mathbb Z\\0&1&y\mathbb ...
3
votes
0answers
52 views
How to solve a distance problem inside of a picture?
sorry for my bad english. I have the following problem:
In the picture you can see 4 different positions. Every position is known to me (longitude, latitude with screen-x and screen-y).
Now i want ...
3
votes
0answers
39 views
Generation of unimodular matrices with bounded elements
Does anybody know what is the algorithm for generating random unimodular matrices, i.e. integer matrices with the determinant +1 or -1 (whose elements do not exceed a given bound)? Such algorithm is ...
3
votes
0answers
68 views
Prove (*) by induction on k.
Challenge: For linear systems with constant coefficients, in some sense we "never need more" than the so-called exponential polynomials, meaning expressions in the form
$$\sum_{i=1}^m ...
3
votes
0answers
61 views
Elementary Lower Matrices
First of all forgive me for my lack of format.
I want to prove that the following elementary lower triangular $nxn$ matrix $Li(x)= I-xe(i)^T$ where $x=[0 \ldots 0 x(i+1) \ldots x(n)]^T$ has the ...
3
votes
0answers
64 views
How do I calculate the circulant determinant $C(1, a, a^2, a^3,\dots , a^{n-1})$?
The question is pretty straight-forward: how do I calculate the circulant determinant $C(1, a, a^2, a^3,\dots , a^{n-1})$ ?
3
votes
0answers
60 views
Jacobi's Rotation has two possibilities, why do they both result in same upper triangular magnitude norm?
The Jacobi's rotation is the complex Givens rotation (unitary similarity) that results in a zero for a specified element of a matrix. If the element
is not adjacent to the diagonal, then there are ...
3
votes
0answers
70 views
Question about linearization
Given a data matrix $D\in\mathbb{R}^{N \times N}$
Can one construct another matrix $M$ that for all permutation matrices $Q^A$,$Q^B$,
if $[\sum_i\sum_j (Q^A_{ij}D_{ij})]^2 \geq [\sum_i\sum_j ...
3
votes
0answers
77 views
Matrix trace minimization and zeros
I would like to minimize and find the zeros of the function
$$F(S,P) = trace(S-SP^{T}(A+ PSP^{T})^{-1}PS)$$
in respect to $S$ and $P$.
$S$ is symmetric square matrix.
$P$ is a rectangular matrix
...
3
votes
0answers
96 views
What kind of matrix/tensor notation is this?
I'm hoping someone on here recognises this and has an answer, because I'm having serious memory issues.
About a year ago, I came across the following way of representing tensors of rank $n$ in matrix ...
3
votes
0answers
42 views
When can the commutator of two matrices be neglected in series expansions?
Under what conditions can the higher order commutators in the Baker–Campbell–Hausdorff formula be neglected when the commutators does not vanish exactly and there is no small parameter in the ...
3
votes
0answers
43 views
finding the largest $p$ components of $x$
Given an $n \times n$ matrix $A$, and an $n \times 1$ vector $b$, the conventional way of computing an $n \times 1$ vector $x$ such that $x=Ax+b$ is to use the following iterations:
...
3
votes
0answers
89 views
Infinite series of matrices almost but not quite matrix exponential
I'm working on a problem that has brought up for me the need to address infinite series of the following form,
$$
\sum_{i=k}^\infty \frac{1}{i!}A^{i-k+1}
$$
where $A$ is an $n\times n$ matrix. If $k = ...
3
votes
0answers
228 views
Good introductory book for matrix calculus
Hi I am an electronics graduate and working on image processing for the past one year...I have a basic exposure to linear algebra(thanks to Gilbert Strang..!!!). Now I am facing problems with matrix ...
