Questions tagged [matrices]
Questions where the notion matrix has an important or crucial role (for the latter, note the tag matrix-theory for potential use). Matrices appear in various parts of mathematics, and this tag is typically combined with other tags to make the general subject clear, such as an appropriate top-level tag ra.rings-and-algebras, co.combinatorics, etc. and other tags that might be applicable. There are also several more specialized tags concerning matrices.
3,106
questions
-5
votes
0
answers
32
views
Chain Rule of Quadratic Matrix? [closed]
Hi I want to find an easy way to get the derivative of this function expressed in matrix format. What is results of df/dy? Thanks a ton!!
χ and y are vectors
Λ and Ω are square matrices
f(y)=(χ-Ωy)'Λ(...
1
vote
1
answer
110
views
Interpreting positive semidefinite matrix as a graph
Given any symmetric matrix $S \in \mathbb{R}^{n \times n}$, if $S \succeq 0$, is there a way to encode $S$ into a graph such that it takes into account the positive semidefinite constraint, and ...
3
votes
1
answer
105
views
The rank of a certain linear combination of mutually commuting nilpotent matrices
Let $A_1,\ldots,A_r$ be mutually commuting $n\times n$ nilpotent matrices over $\mathbb C$, the field of complex numbers. For any complex number $c$, let $A(c):=A_0+cA_1+c^2A_2+\ldots +c^rA_r$. We ...
-1
votes
0
answers
34
views
simpler question from sylvesters eqn [closed]
i think i can simplify my problem from sylvesters
equation (AX - XB = C) to this:
(D_1) X - X(D_2) = 0
where
D_1 is diagonal nxn
D_2 is diagonal mxm
X is nxm
D and D_2 are diagonal matrices
and they ...
2
votes
0
answers
44
views
Relationship between the homology of two types of tensor products of $\mathbb{Z}/ 2 \mathbb{Z}$-graded objects?
Let's consider a $2$-periodic complex $F$ of free $R$-modules, which is just a $\mathbb{Z} / 2 \mathbb{Z}$-graded complex
$$F_1 \xrightarrow{d_1} F_0 \xrightarrow{d_0} F_1$$
(really the arrow $d_0$ ...
1
vote
1
answer
33
views
Iteration matrix representation with complex conjugate operator
I am studying the convergence of a particular class of radial power flows, whose goal is to obtain the voltage solution for a given electric grid, i.e., a complex vector $\mathbf{V}$ that gives the ...
0
votes
1
answer
72
views
Every square complex matrix is similar to a TRIDIAGONAL complex-symmetric matrix?
Every square complex matrix is similar to a complex-symmetric matrix. But I think a stronger statement is also true: Every such matrix is also similar to a TRIDIAGONAL complex-symmetric matrix. Where ...
0
votes
1
answer
146
views
“Smallest” non-zero linear combination of vectors to obtain a non-negative vector
We say that a vector $\mathbf{x}$ in $\mathbb{Z}^j$ is non-negative if it is of the form
\begin{bmatrix}
x_1 \\
x_2 \\
\vdots \\
x_j \\
\end{bmatrix}
where $x_{j} \geq 0$.
Suppose that ...
2
votes
1
answer
250
views
Continuity of eigenvector of zero eigenvalue
Wonder whether anyone has an idea on showing the following or to point out that it is not true:
Let $A(t) \in \Re^{n \times n}$ be differentiable over an interval $I$, and it has a zero eigenvalue for ...
3
votes
2
answers
376
views
Monotonicity of matrix conjugation
Let $A$ and $B$ be positive-definite matrices such that $A \le B.$
By matrix monotonicity of the root, this also implies that $A^{\alpha} \le B^{\alpha}$ for $\alpha \in [0,1].$
I am now curious under ...
1
vote
2
answers
112
views
Property for bounding matrix exponential
Wikipedia states in the exponential map section about the exponential of a matrix that for any matrices $X$, $Y$ it holds that $\|e^{X+Y}-e^{X}\| \leq \|Y\|e^{\|X\|} e^{\|Y\|}$ where $\|\cdot\|$ ...
-1
votes
0
answers
77
views
How to find two sets of vectors which satisfy a set of matrix equations [migrated]
In my trial to solve a system of matrix equations, I wish to find two sets of non-zero vectors of $\mathbb{R}^3$ (which may be not unique) $\{ A_i \}$ and $\{ B_i \}$ where $i \in I$ (an index set, ...
3
votes
0
answers
47
views
About a circular variant of Vandermonde matrix
Given an arbitrary $(x_1, \dots, x_n) \in [0, 1]^n$, is there any name/known results for the following $n \times n$ matrix (which is constructed by iterating $(x_1 \to \dots \to x_n \to x_1 \to \dots)$...
4
votes
2
answers
121
views
What does the matrix-valued solution of $X = A X A^T + \operatorname{Id}$ look like?
Let $A \in \mathbb{R}^{n \times n}$ be an invertible contraction, i.e. all singular values are in $(0,1)$. By reformulating the equation
\begin{align*}
& X = A X A^T + \operatorname{Id} \tag{1}
\...
6
votes
1
answer
373
views
Arithmetic properties of positively reduced $2\times 2$-matrices
Call a $2\times 2$ matrix with coefficients in $\{0,1,2,3,\ldots\}$
positively reduced if any row or column reduction (given by replacing a row/column by itself minus the other row/column) produces
at ...
2
votes
1
answer
294
views
$\omega$-Commuting matrices vs Stone-von Neumann Theorem
Let me first recall the Stone-von Neumann theorem that if two one-parameter groups of unitary operators $U_t$ and $V_s$ over a Hilbert space satisfy $U_tV_s=e^{ist}V_sU_t$ for every $s,t\in{\mathbb R}$...
126
votes
13
answers
26k
views
Should the formula for the inverse of a 2x2 matrix be obvious?
As every MO user knows, and can easily prove, the inverse of the matrix $\begin{pmatrix} a & b \\\ c & d \end{pmatrix}$ is $\dfrac{1}{ad - bc} \begin{pmatrix} d & -b \\ -c & a \end{...
1
vote
0
answers
18
views
Correlation Matrix Problem of Three Decomposition Level of DWT
I'm trying to apply a DWT with 3 composition levels and the following question arose when calculating the composition matrix.
The step I'm trying to follow is:
The DWT coefficientes are obtained from ...
5
votes
2
answers
337
views
Existence of a specific stochastic matrix
Let $0\le x_1\le x_2\le \cdots\le x_n\le n-1$ be given. My question is as follows : Under which condition there exists a doubly stochastic matrix $M=(m_{i,j})_{1\le i,j\le n}$ s.t.
$$\sum_{j=1}^n (j-1)...
0
votes
0
answers
34
views
some problem about the discrete of the first derivative operator
I am reading a paper
(Parameter Choice Strategies for Multipenalty Regularization Massimo Fornasier, Valeriya Naumova, and Sergei V. Pereverzyev SIAM Journal on Numerical Analysis 2014 52:4, 1770-1794)...
0
votes
2
answers
57
views
Nondegeneracy of dominant singular value and positivity of dominant singular vector of connected nonnegative matrix
Call a (not necessarily square) nonnegative matrix $M$ connected if there do not exist permutation matrices $P$ and $Q$ such that $PMQ=\begin{pmatrix}A&0\\0&B\end{pmatrix}$ for some $A$ and $B$...
3
votes
0
answers
73
views
A stochastic matrix $B = \lambda(\lambda I - A)^{-1}$ such that $B-B^2$ has a non-negative diagonal
I apologize if this is too elementary a question, but I have not been able to make much progress.
Consider a real matrix $A$ with $A_{ij} >0$ for $i \ne j$ and $\sum_{j} A_{ij} = 0$ for each $j$. ...
0
votes
1
answer
61
views
Handling the $\ell^2$ norm of a matrix expression in a linear regression
I am reading a scientific article in which matrices are handled (which I do not use often). We consider a matrix $X\in\mathbb R^{n\times p}$ and a vector $y\in\mathbb R^n$. The authors show that the ...
0
votes
0
answers
42
views
Find condition on signed matrix $A$ such that $\sum_jA_{ij}\cos(\theta_i-\theta_j)\geq 0$ implies $\sum_j\cos(\theta_i-\theta_j)\geq 0$?
As stated in the title, My question is:
Suppose $\sum_jA_{ij}\sin(\theta_i-\theta_j)=0,\forall i\in[n]$.
How to find condition on symmetric, diagonal-free, signed matrix $A$ ($A_{ij}=+1$ or $-1$ if $i\...
6
votes
0
answers
187
views
Zero-one pairings between sets of vectors
Let
$A\subseteq V$ and
$B\subseteq V^\star$
be spanning sets in
a finite-dimensional real vector space $V$ and
its dual $V^\star$.
Suppose that
$$
\langle b,a\rangle\in\lbrace0,1\rbrace
$$
for all
$a\...
2
votes
1
answer
141
views
Upper bound for the rank of a Gram-type matrix
Let $V = (v_{1},...,v_{N})$ and $W = (w_{1},...,w_{N})$ be 2 sets, each containing $N$ vectors from $\mathbb{R}^{n}$; i.e, $v_{j}, w_{j} \in \mathbb{R}^{n}$ for all $1 \leq j \leq N$. Assume that $N$ ...
1
vote
2
answers
144
views
Right inverse of integer matrix
If I have a rectangular matrix $A$ (say $4 \times 6$) with integer entries, is there a way to tell whether it has a right inverse that also has integer entries. I know that if $AA^T$ has determinant $...
10
votes
3
answers
2k
views
Trace inequality for non-reversible Markov chain
Let $P \in \mathbb{R}^{d \times d}$ be the transition kernel for a Markov chain with stationary measure $\pi$ and define $P^\ast$ to be the time-reversed transition kernel defined by $P^\ast_{ij} := ...
-2
votes
1
answer
130
views
Norm of a matrix function of a vector $x$ [closed]
I have a matrix $$A(x) = \frac{-1}{(1+\|x\|_2^2)^{\frac{3}{2}}}xx^T + \frac{1}{(1+\|x\|_2^2)^{\frac{1}{2}}}I$$ I see that $$\|A(x)\|_2 \le 1 \ \forall x$$
But is $\|A(x)\| \le 1$ in general $\forall x$...
-2
votes
3
answers
2k
views
Convergence of a markov matrix
Consider a markov chain matrix P of size n x n (n states).
P is known to be:
1- Not irreducible (i.e. there exist at least a pair of states i, j such that we cannot go from i to j)
2- Not all ...
6
votes
2
answers
975
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 ...
3
votes
1
answer
222
views
Cofactor matrices and positive semi-definiteness
If $A$ is an $n\times n$ matrix with real entries, let me write $\widehat A$ its cofactor matrix. Since the map $A\mapsto\widehat A$ is polynomial, homogeneous of degree $n-1$, it can be multi-...
0
votes
0
answers
68
views
Quick calculation of a symmetric product with two indices
Say I have a product $\prod_{1\le i \le N-1}\prod_{i<j\le N-1} (1+t_i t_j a_{ij})$, where $a_{ij}$s are real number. I want to calculate the coefficient of $\prod_{0 \le i < N} t_i$. Is there an ...
5
votes
1
answer
212
views
Are there any applications of the algebraic polar decomposition?
One of the decompositions mentioned in the Wikipedia page on matrix decompositions is the algebraic polar decomposition. This factors a square complex matrix $M$ into $M = SQ$ where $S = S^T$ and $QQ^...
6
votes
3
answers
573
views
Reference for partial Hadamard matrices
Definition. An $m\times n$ matrix is said to be a partial Hadamard matrix (let's say PHM) if its entries are chosen from $\lbrace -1, 1 \rbrace$ such that the dot product of each pair of row vectors ...
1
vote
1
answer
165
views
Matrices over a finite field: matrices for which some unipotent $U$ satisfies Trace$(ZU)=0$ for all $Z$ in the commutant
Let $p$ be an odd prime number, let $A\in M_p(\mathbb{F}_p)$ be a $p$-by-$p$ matrix with coefficients in $\mathbb{F}_p$, let $C(A)$ be the commutant of $A$, and let $N\in M_p(\mathbb{F}_p)$ be a ...
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}
$$
where $A,B,C,D$ are diagonalizable square matrices and I want to ...
4
votes
1
answer
193
views
A question on eigenvalue of parametric matrix
Is there a way to efficiently check if all matrices in the following set are Hurwitz stable (eigenvalues strictly in the left-hand plane)?$$\left\{ A \in \Bbb R^{n \times n} : \ell_{i,j}\leq A_{i,j} \...
0
votes
1
answer
235
views
Eigenvalues of $\operatorname{diag}({\bf v}) - {\bf v} {\bf v}^\top - \alpha({\bf v} - {\bf w})({\bf v} - {\bf w})^\top$
Given vectors ${\bf v}, {\bf w} \in [0,1]^n$ , where $n \in \mathbb{N} \setminus \{0\}$, and $\alpha > 0$, I would like to find the eigenvalues of the following matrix.
$$\operatorname{diag}({\bf v}...
0
votes
1
answer
134
views
Full matrix ring over an infinite division ring with a finite maximal unital subring?
I'm wondering if there is an infinite division ring $D$ and a finite unital subring $R$ of the full matrix ring $M_n(D)$ ($n$ some positive integer) such that there are no rings properly between $R$ ...
20
votes
2
answers
1k
views
Find $Y\in\operatorname{GL}_n(\mathbb{Z})$ such that all eigenvalues of $YX$ are nonnegative
I saw this problem some years ago and I would greatly appreciate any reference or solution.
Let $X \in \operatorname{M}_n ( \mathbb{R} )$. Prove that there is $Y \in \operatorname{M}_n ( \mathbb{Z} )$...
1
vote
1
answer
102
views
Solution to $a=e^t (t-r_1)(t-r_2)$ with Lambert $W$ function, where $r_1, r_2 $ are complex
Lambert $W$ works when $r_1$, and $r_2$ are real. However, I am trying to solve the equation when $r_1$, and $r_2$ are complex numbers.
1
vote
1
answer
330
views
How to represent infinite matrices in Mathematica?
I asked this question on Mathematica site and the answer was basically "it is impossible". So, some math theory needs to be thrown in besides coding.
Let us represent infinite matrices as ...
2
votes
1
answer
131
views
Generate a low-rank sparse covariance matrix
May I ask how to generate a low-rank sparse covariance matrix? Thank you!
2
votes
1
answer
414
views
Integrality certification for product of two matrices $A B^{-1}$
Let's consider two non-singular integer matrices $A,B \in\mathbb{Z}^{n\times n}$. I want a test to check if $A\times B^{-1}$ is integral (or no denominators). I am referring the unimodular ...
3
votes
0
answers
150
views
What are the measure of the volume and boundary (and other quermaß measures) of the positive semidefinite matrices?
Let $E$ be the real vector space of $n\times n$ real symmetric (resp. complex Hermitian) matrices, and $E_1$ those with trace $1$. Endow $E$ with the bilinear (resp. sesquilinear) form given by $(P,Q)...
1
vote
1
answer
332
views
Solvability of $A X B=C$ with $X=X^\mathrm{T}$
I am studying symmetric solutions to the complex matrix equation
\begin{equation}
A X B=C,
\end{equation}
where $A$, $B$, and $C$ are $m\times n$, $n \times k$, and $m \times k$ complex matrices, ...
0
votes
0
answers
32
views
What is the impact of individual estimate on each other in matrix inversion?
I am looking to understand the impact of each estimate on each other in matrix inversion.
Lets say I have a vector $A = \left[a_1, a_2 \right]^T$ of size $2 \times 1$ and $a_1$ and $a_2$ are related ...
2
votes
1
answer
418
views
Effect of duplicated row on singular values and vectors
Let $\mathbf{A}$ be a $n\times n$ matrix with Singular Value Decomposition (SVD) $\mathbf{A}=\mathbf{U}\mathbf{S}\mathbf{V}$ and $\mathbf{a}_1$ be the first row of $\mathbf{A}$. What can we say about ...
3
votes
1
answer
373
views
Minimum upper bound for sum of the entries of the inverse covariance matrix
Let $x \in \mathbb{R}^n$ and $k$ is RBF kernel
$$k(x, x') := \exp \left(-\frac{\|x-x'\|^2}{2\sigma^2}\right)$$
and let $\mathbf{K}$ be the following $n \times n$ covariance matrix
$$\mathbf{K} = \...