All Questions
Tagged with matrices matrix-theory
248 questions
1
vote
0
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137
views
Boundary of pseudospectra
Suppose:
$B_i \in \mathbb{C}^{n \times n}$, $0<w_i\in \mathbb{R}$ $(i = 0,1,2,\ldots,m)$
${\rm P}(x) ={\rm{B}_m} x ^m + \cdots + B_1 x + B_0$ is a matrix polynomial, and $x $ is a complex ...
- 151
1
vote
1
answer
940
views
Uniqueness and invariance of the LDLT decomposition
A real symmetric positive semi-definite matrix $A$ can be decomposed in the form
$A = P^TLDL^TP$,
where $P$ is a permutation matrix, $L$ is a lower unit triangular matrix and $D$ is a diagonal ...
- 101
8
votes
2
answers
12k
views
Relation between eigenvalues of $A$ and $A^TA$?
For an $n\times n$ diagonizable matrix $A$, is there a relation between the eigenvalues of $A$ and the eigenvalues of $A^TA$?
I ask this because I am looking into the relation between $A$ and $A+cI$, ...
- 181
13
votes
2
answers
6k
views
Parametrization of positive semidefinite matrices
We know that a real, symmetric, positive definite matrix $A$ of size $n\times n$ can be parametrized by a vector $\theta$ of $\frac{n(n+1)}{2}$ parameters thanks to the Cholesky decomposition:
$$
A = ...
- 313
-1
votes
1
answer
230
views
How bad could $\|A^k\|$ be when $\rho(A) < 1-\delta$ [closed]
(Sorry, I do hate editing this many many times but let me try the last time)
Gelfand's formula says that
$$\lim_{k\rightarrow \infty} \|A^k\|^{1/k} = \rho(A)$$
I am wondering whether there is any ...
- 181
8
votes
1
answer
2k
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"Additive version" of Kronecker product
Let $A$ and $B$ be two square matrices with complex entries.
Let $\lambda_1, \ldots, ,\lambda_n$ be the Eigenvalues of $A$ and
$\mu_1, \ldots, ,\mu_m$ be the Eigenvalues of $B$.
Then the Eigenvalues ...
- 3,031
15
votes
1
answer
782
views
Is SL(n,Z[x]) generated by transvections?
Is $\mathrm{SL}(n,\mathbb{Z}[x])$ equal to $E(n,\mathbb{Z}[x])$, the subgroup generated by transvections?
- 281
0
votes
0
answers
160
views
$l_{\infty}$ norms of matrix perturbations
Say $B$ is a real symmetric matrix of dimension $n$ and $A$ is another real symmetric matrix of the same dimension.
What needs to be the bounds on (which?) norm of $B$ to ensure that $\lambda_{max}(...
- 1,893
9
votes
3
answers
544
views
Product of a Finite Number of Matrices Related to Roots of Unity
Does anyone have an idea how to prove the following identity?
$$
\mathop{\mathrm{Tr}}\left(\prod_{j=0}^{n-1}\begin{pmatrix}
x^{-2j} & -x^{2j+1} \\
1 & 0
\end{pmatrix}\right)=
\begin{cases}
...
- 309
9
votes
2
answers
1k
views
On closest unitary matrix
In this question $\|A\|_p$ is the normalized $p$-th Schatten norm which is defined to be $\left(\mathbb E_{i} \lambda_i^p\right)^{1/p}$, where $\lambda_i$ are singular values of matrix $A$.
Suppose ...
- 901
1
vote
0
answers
113
views
Is my particular finite dimension Toeplitz matrix always strictly positive?
Let $H(\omega), \; -\pi \leq \omega \leq \pi$ be a real-valued function with a continuous band of zeros, that is (for simplicity) $H(\omega)=0, \; |\omega|\geq \beta \pi$.
Define a sequence of banded ...
- 21
13
votes
1
answer
275
views
What is the symmetry group fixing norms of elements of a unitary matrix?
Let $N\geq1$ be an integer and let us say that two matrices $U,V\in U(N)$ are related if $|U_{ij}|=|V_{ij}|$ for all indices $1\leq i,j\leq N$.
When exactly are two unitary matrices related in this ...
- 8,086
3
votes
2
answers
522
views
quadratic matrix equation
Find all symmetric matrices $X=X^{T}$ such that
\begin{align}
XDX^{T}=-D \quad (1)
\end{align}
where $D\ne 0$ is a real diagonal matrix.
For example, $X=iI$ satisfies $(1)$. Can you get a ...
- 75
5
votes
0
answers
435
views
When is a Hankel matrix invertible?
I would like to have some conditions that render a general Hankel matrix of the form
\begin{pmatrix}a_1 & a_2 & a_3 &\cdots & a_n \\ a_2 & a_3 & a_4 & \cdots & a_{n+1} \...
- 327
2
votes
0
answers
677
views
Bounds on smallest Eigenvalue of the Sum of a Standard Laplacian and a Diagonal Matrix
I'm trying to find upper boundaries on the smallest Eigenvalue $\lambda_1$ of $L + E$, where $L$ is a standard Laplacian of an unweighted digraph, with $\lambda_1(L) = 0$ and $E \in \{0,1\}^{n \times ...
- 21
4
votes
1
answer
262
views
Calculating the dimension of the algebra generated by some given matrices
Let $X_1, X_2, \ldots, X_d$ be $n \times n$ matrices over some field $K.$ I want to calculate the dimension of the unital algebra generated by $X_1, X_2, \ldots, X_d$ for some examples in a problem I ...
- 1,429
3
votes
0
answers
611
views
Can anyone help me deduce a matrix inequality?
The following lemma is taken from references firstly.
Lemma 1 [1-2] Given matrices $Q=Q^{T} , F, M$ and $N$ of appropriate dimensions, then $$Q+MFN+N^{T}F^{T}M^{T}<0$$
for all $F$ satisfying $F^{...
- 103
0
votes
1
answer
546
views
Solution of infinite dimension linear system
Suppose that ${a_n}$ and $b_n$ is decreasing sequence such that $a_0=A$, $lim_{n->\infty}a_n=0$ and $b_0=B$, $lim_{n->\infty}b_n=0$.
For fix n,
we can construct n dimension linear equation ...
- 111
11
votes
1
answer
453
views
A variant of Cholesky decomposition involving binary matrices
Studying a problem that is not directly related to linear algebra I came across the following problem.
Let $B$ be $n \times n$ symmetric matrix whose entries are non-negative integers. I would like ...
- 3,463
5
votes
2
answers
322
views
Stabilization of the pencil of skew symmetric matrices by the orthogonal group
During my researches I've come across the following question.
Let $A$ and $B$ be a couple of square $k\times k$ skew symmetric matrices on $\mathbb R$. Let us consider the (real) pencil generated by $...
user17697
21
votes
3
answers
51k
views
What is the time complexity of truncated SVD?
Full SVD, on an $m \times n$ matrix $A$, [U,S,V] = svd(A), would cost $O(m^2n + mn^2 + n^3)$ time. But what is the time complexity if we only need the $k$ largest ...
- 327
1
vote
0
answers
305
views
how to find all the solutions to $I+A+\cdots+A^n=0.$ [closed]
Let $GL_3(\mathbb{Z}[i])$ be the group of invertible $3\times 3$ matrices whose coefficients are Gaussian integers.I want to find all the pair $(A\in GL_3(\mathbb{Z}[i]),n\in\mathbb{Z})$ satisfying
$$...
- 197
0
votes
1
answer
666
views
Kernel of a projection
Given $m<n$. Suppose that $H$ and $K$ be $m \times n$ and $n\times (n-m)$ matrices such that rank$(H)=m$, rank$(K)=n-m$, and $HK=0$. For fixed non singular symmetric matrix $A$ define
\begin{...
- 133
1
vote
1
answer
517
views
Books or references on multidimensional matrix operations [closed]
Have the 2D matrix operations been generalized to n-dimensional matrices?
Are there any books that define various operations on multidimensional matrix? I'd like to see operations such as ...
- 163
1
vote
1
answer
18k
views
Derivative of log determinant and inverse
I have a matrix $\Sigma$ with element $(i,j)$
$$\Sigma_{i,j}= \exp(-h_{i,j}\rho).$$
The matrix is positive definite and symmetric (it is a covariance matrix).
Now I need to evaluate
$$\frac{\...
- 29
2
votes
0
answers
132
views
Characterizing the singular values of a matrix with structure
Suppose we have a function from $\mathbb{R}^2\to\mathbb{C}$,
$$f(x,y) = e^{\imath\pi x g(y)}$$
where $g(y)$ is periodic in $y\in[-T, T),\ T<\infty$ (e.g., a sinusoid) and $0\leq x < \infty$
...
- 21
23
votes
0
answers
8k
views
An $n \times n$ matrix $A$ is similar to its transpose $A^{\top}$: elementary proof?
A famous result in linear algebra is the following.
An $n \times n$ matrix $A$ over a field $\mathbb{F}$ is similar to its transpose $A^T$.
I know one proof using the Smith Normal Form (SNF). ...
- 3,320
2
votes
2
answers
421
views
On matrix norms
It is standard to define an induced matrix norm $|||\cdot|||$ from a vector norm $||\cdot||$ in this way:
$|||A|||=\max_{x \neq 0}{\frac{||Ax||}{||x||}}$.
Suppose we define a different function of ...
- 6,962
4
votes
2
answers
372
views
system of homogeneous matrix equations
Edit: Maybe I should make it clear that by a solution I mean a pair of matrices $(A,B)$ such that the identity below holds for any complex numbers $x,y$.
One of my friend asked me the following ...
- 411
7
votes
2
answers
1k
views
Matrix groups and presentation
Suppose $K$ is a number field and I have a subgroup of $\operatorname{GL}_2(K)$ for which I know a (finite) set of generators. Is there an algorithm that gives me a presentation of the group?
More ...
- 1,271
17
votes
1
answer
3k
views
2x2 subdeterminants of a matrix
If N>2, it is well known that if two invertible NxN matrices A and B have the same determinants of any 2x2 corresponding submatrices, then A=B or A=-B.
Given then all these 2x2 determinants of an ...
- 1,309
1
vote
1
answer
206
views
What is such an equation called?
Is there a name and common technique for such equations, where $A$ and $B$ are matrices and $x$ a vector?
$Ax+f(\lambda)Bx=g(\lambda)x$.
- 6,962
2
votes
1
answer
276
views
Asymptotic Behavior of Non-Analytic Function of the Eigenvalues
Hello,
Let $A_n = (a_{k-j};\;k,j = 0,1,\ldots,n-1)$ be a sequence of $n\times n$ Toeplitz matrices, with eigenvalues $(\lambda_{n,i};\;i = 0,1,\ldots,n-1)$.
If $A_n$ were a sequence of Hermitian ...
- 43
2
votes
1
answer
2k
views
power of a block triangular matrix
I have a matrix in the form :
$$M =
\begin{pmatrix}
A & 0 & 0 \\\
B & A & 0 \\\
C & D & A
\end{pmatrix}
$$
where $A,B,C,D$ are diagonalizable square matrices and I want to ...
- 21
4
votes
2
answers
5k
views
Estimating a spectral gap
Suppose you have a real positive definite matrix $A$ who eigenvalues are $\lambda_{1} \leq \lambda_{2} \leq \ldots \leq \lambda_{n}$. I am interested in bounding from below $\lambda_{2}-\lambda_{1}$. (...
- 6,962
1
vote
3
answers
5k
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Number of parameters needed to specify a Hermitian matrix of rank r.
Hi,
i have two questions that seem to bother me lately. Maybe you could help me and/or point me in any related literature.
1) Assume hermitian matrix $H \in \mathcal{C}^{n \times n}$ that has rank $...
- 199
2
votes
1
answer
714
views
Is there a natural distance between skew hermitian matrices?
Working in machine learning, I try to find a way to compare time series, which can be considered as semi-continuous matrices belonging to $\mathbb R^{n \times \mathbb R}$ (a column corresponds to n-...
- 103
1
vote
0
answers
72
views
sharper interlacing
The usual interlacing inequalities say that if $M$ is a Hermitian $n \times n$ matrix and $\hat{M}$ is a principal submatrix of order $n-1$, then $\lambda_{\min}(M) \leq \lambda_{\min}(\hat{M})$. I ...
- 6,962
5
votes
2
answers
1k
views
Generalizations of Oppenheim's inequality
The well-known Oppenheim inequality says that for two positive definite matrices $A,B$ it holds that $\det(A \circ B) \geq (\prod{a_{ii}})\det(B)$.
There has been a lot of beautiful work done ...
- 6,962
3
votes
2
answers
557
views
S-matrix conjecture: status?
Is the $S$-matrix conjecture still open? I mean the one listed as Problem 7 in this survey.
- 6,962
20
votes
1
answer
25k
views
When does the $4\times 4$ 'false Sarrus rule' compute the determinant correctly?
This question is most probably not research level, but I thought that the MO folks might like it... Feel free to close.
Here is the motivation: If you have ever taught a maths course for engineers ...
- 12.7k
1
vote
1
answer
212
views
name for a matrix operation
If $A$ is a matrix and $D$ is a diagonal matrix, is there some special name for $DAD$?
- 6,962
3
votes
2
answers
2k
views
matrices whose entries sum to zero
Let $A$ be a non-singular matrix and let $s(A)$ be the sum of its entries. Under which conditions can it be assured that $s(A) \neq 0$?
if you like, you can assume that $A$ is symmetric.
Here is an ...
- 6,962
5
votes
2
answers
780
views
A question about matrices with more details
Let $A,B\in\mathbb{R}^{n\times n}$. Suppose that $B$ is nonsingular and that there exists $m$ reals pairwise distinct $\lambda_{1},\cdots,\lambda_{r},\cdots,\lambda_{m}$ such that
$$B^{-1}A=\...
6
votes
0
answers
514
views
concentration for eigenvectors
I am interested in bounding from above the ratio between the maximum and minimum entries of a Perron vector. The only results that I found in the literature are from the classic masters (Ostrowski and ...
- 6,962
1
vote
2
answers
271
views
small sums of entries in submatrices - strange phenomenon
Suppose that $x \in \mathbb{R}^{n}$ is a vector of small positive fractions, i.e. $x_{i} \approx \frac{1}{n}$. The exact values are unknown. I form the matrix $M=diag(x)-\frac{xx^{T}}{2}$ which is a ...
- 6,962
3
votes
1
answer
290
views
Bandwidth reduction of multiple matrices
Suppose I have a symmetric matrix $A$ and several diagonal matrices $D_1,D_2,\dots$. Are there any matrix transformations such as $P^\top A P$ so that
$$P^\top A P, P^\top D_1 P, P^\top D_2 P, \...
22
votes
2
answers
14k
views
Infinite matrices and the concept of "determinant"
Suppose we have an infinite matrix A = (aij) (i, j positive integers). What is the "right" definition of determinant of such a matrix? (Or does such a notion even exist?) Of course, I don't ...
- 1,861