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.
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Upper bound on largest eigenvalue of a real symmetric $n \times n$ matrix with all main diagonal entries positive, everywhere else nonpositive
Is there a good analytic upper bound on the largest eigenvalue of a real symmetric n*n matrix with all main diagonal entries strictly positive, all other entries <=0 with typically many of them ...
2
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1
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Does there always exist a matrix satisfying certain tracial conditions
Given odd integers $0<a<b$, I want to know if there exists an $n$ by $n$ real valued square matrix $M$ such that
$$ M_{ij} = M_{ji} \quad \forall i,j \in \{1,2\dots n\}$$
$$ \sum_{i=1}^n M_{ij} =...
CommunityBot
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Symmetric indefinite matrix of fixed rank — manifold structure?
I have been studying symmetric indefinite matrices of fixed rank, which have been rather useful for a particular application. I wonder if there is a way to parameterise these by a smooth manifold, e.g....
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Change of the smallest positive eigenvalue after a rank-one update
Given natural numbers $n,r,R\in\mathbb{N}$ with $r,R\le n$, let $A\in\mathbb{R}^{n\times r}$ and $B\in \mathbb{R}^{n\times R}$ be two matrices with full column rank and let $c\in\mathbb{R}^n$.
Denote ...
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3
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Which doubly stochastic matrices can be written as products of pairwise averaging matrices?
A matrix $A$ is called doubly stochastic if its entries are nonnegative, and if all of its rows and columns add up to $1$. A subset of doubly stochastic matrices is the set of pairwise averaging ...
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0
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Elliptic operators with Robin boundary conditions
Can it be proved that two elliptic operators with Robin boundary conditions generate an interval $P$-matrix?
$$
-a\Delta u_i = f_i, \quad a\frac{\partial u_i}{\partial n} + bu_i = 0
$$
$$
-a\Delta v_i ...
- 243
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1
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Necessary and sufficient conditions for a sum of idempotents to be idempotent
Given: a finite list of $n$-by-$n$ idempotent complex matrices $E_1, E_2, \ldots, E_k$.
If all pairwise products $E_i E_j$ (with $i \neq j$) are zero, it is trivial to show the sum $E_1 + E_2 + \cdots ...
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Determinant of an updated Covariance matrix
I am faced with the following problem :
Originally (at time $0$) I have a number of data samples $x^0_{1,\ldots,n}$ (normalised : $E[x] = 0, \operatorname{Var}[x] = 1$) from which I have calculated ...
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Determinant of the random matrix $X^2+Y^2$
$\DeclareMathOperator\Prob{Prob}$Let $X,Y\in M_n(\mathbb{R})$ be $2$ random matrices. The entries of $X,Y$ are i.i.d. variables. They follow the standard normal law $N(0,1)$.
i) When $n=2,3,4$, one ...
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1
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Continuous path of unitary matrices with prescribed first column?
Consider a continuous curve $u \colon [0,1] \to \mathbb{C}^n$ where $u(t)$ is always a unit vector, $u(t)^* u(t) = 1$.
Question 1: Does there exist a continuous curve $U \colon [0,1] \to \mathbb{C}^{n ...
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Does this matrix equation always have a solution?
Let $\{A_i, i\ge 3\}$ be the matrices whose columns represent numbers from $0$ to $2^i-1$ in binary form. For example,
$A_3 = \begin{bmatrix} 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 \...
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Expected value of the largest singular value of a random matrix with entries in $N (0,1)$
Given a matrix $A \in \mathbb R^{n \times n}$ whose entries are i.i.d. $N(0,1)$, what is the expected value of its largest singular value? Equivalently, what is the expected value of the largest ...
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Wold decomposition of toral endomorphisms
Suppose that $A\in M_d(\mathbb{Z})$ is a $d \times d$ matrix with non zero determinant and suppose that $\mathbb{T}^d$ is the $d$-dimensional torus. Then one can define an operator on $L^2(\mathbb{T}^...
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Low-rank factorization of a Finite Element matrix
I have a matrix $M\in \mathbb R^{n\times n}$, stemming from a Finite Element discretization of an advection function.
I want to find a factorization $ M= S E T $ with $S, T\in \mathbb R^{n\times s}$ ...
3
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1
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Calculate the Riemannian Hessian of Karcher mean problem on positive definite matrices
Consider a collection of positive definite matrices $\{A_1,...,A_n\}\in\mathbb{S}_{++}^d$, the Karcher mean of these matrices is given by (see (5.4) in [1]):
$$
\min_{X\in\mathbb{S}_{++}^d} f(X):=\...
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Is there a conceptual reason why every square complex matrix is similar to a complex-symmetric matrix?
The question is maybe a bit vague, but like the title says: Every square complex-matrix $M$ is equal to $P S P^{-1}$ where $S = S^T$. The proof begins by taking the Jordan Normal Form of $M$, and then ...
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Eigenvalues of a subset of matrix semigroup
My apologies for slightly longer post but I wanted to explain lower dimensional cases and their proofs before asking the actual question, which starts after the phrase The general case below.
A two-...
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Non-vanishing principal minors up to swapping columns
An undergraduate student asked me the following seemingly easy question. After a few days of thinking, I still couldn't come up with an answer, nor could I find one online. Maybe folks here could help?...
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Are these two methods for constructing Hadamard matrices known?
These two observations came while researching the empty set of odd perfect numbers and unitary perfect numbers:
Context:
Let $n$ be a natural number and $D_n$ be the set of divisors.
We can make this ...
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Minimum distance of a symmetric matrix to diagonal matrices
Let $A=(a_{ij})$ be an arbitrary $n \times n$ real symmetric matrix and $n\geq 2$. Let $\| \cdot \|$ denote the operator $2$-norm or equivalently the maximum absolute value of eigenvalues for ...
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Pre-positive definite functions?
A function $f(x,y)$ is positive definite if matrices $( f(x_i, x_j) )_{i, j \in F}$ are positive definite for all finite index sets $F$. This is frequently hard or impossible to check given some ...
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Existence of cyclic subspace decompositions for pairs of commuting matrices
Let $\mathbb{K}$ be an arbitrary field (possibly finite). Let $V$ be a finite-dimensional vector space over $\mathbb{K}$, and let $A,B$ be two linear endomorphisms of $V$ which commute.
For $v\in V$, ...
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Real square roots of symmetric matrices
In joint work with Andreas Fischle (TU Dresden, Germany) and Patrizio Neff (U Essen, Germany) we needed to use the following statement: If $S$ is a real $n\times n$ matrix with $S^2$ symmetric, then ...
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Inductive Cholesky decomposition to prove that a function is positive definite over the natural numbers?
I am trying to prove that the function:
$$k(a,b):=\frac{1}{\operatorname{rad}\left ( \frac{ab}{\gcd(a,b)^2} \right )}$$
is a positive definite function over the natural numbers. What has sometimes ...
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Solve permutation matrix equations of the form: $X^T A X = B_1$ and $X A X^T = B_2$
I have a hard time solving the following two matrix equations for unknown permutation matrix $X \in \mathbb{R}^{n \times n}$:
$$X^T A X = B_1$$
$$X A X^T = B_2$$
where, $A$, $B_1$ and $B_2$ are all $n ...
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Jensen-like inequality for random matrix: $\Bbb E[\det X^2]\ge\det\Bbb E[X^2]$
Let $X\in M_n(\Bbb R)$ be a random matrix with iid elements following a continuous distribution.
What are the necessary and sufficient conditions for $$\Bbb E[\det X^2]\ge\det\Bbb E[X^2]$$ to hold? Is ...
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Is there a combinatorial interpretation for the change of basis matrix in the Frobenius normal form representation?
Let $G$ be a graph on $n$ vertices. Let $A$ be the adjacency matrix of $G$ (i.e., rows and columns of $A$ are indexed by vertices of $G$, and the $(v,w)$ entry of $A$ is $1$ if $(v,w)$ is an edge in $...
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Is Hilbert–Schmidt and Frobenius norm the same?
From the definition on $\Bbb R$ those two norm are the same:
the Frobenius norm,
the Hilbert-Schmidt norm.
Is there some difference (on $\Bbb C$) or historical reason for two names for the same ...
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Matrix valued word embeddings for natural language processing
In natural language processing, an area of machine learning, one would like to represent words as objects that can easily be understood and manipulated using machine learning. A word embedding is a ...
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Is there a necessary and sufficient condition to determine whether a number sequence can serve as the first few moments of a Radon measure?
Given a few positive numbers $(M_1, M_2,\cdots, M_K)$, they are the moments of a measure if
\begin{equation}
M_k = \int d\mu(x) x^k,\quad k = 1,2,\cdots,K.
\end{equation}
This is related to the ...
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An inequality related to matrix trace
$$Tr(A|A^T U \Sigma U^T|) \leq Tr(AA^T U \Sigma U^T)$$ where $A \in \mathbb{R}^{m\times n}$ is a real rectangular matrix, $U \in \mathbb{R}^{n \times n}$ is a orthonormal matrix and $\Sigma$ is a ...
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Uniquely reconstruct a matrix $M$ from its inverse $M^{-1}$ if $n$ elements of $M^{-1}$ are unknown and $n$ elements of $M$ are given
This question was motivated by a recent MO post. You know $n$ elements of the $N\times N$ matrix $M$ and you do not know $n$ elements of the inverse $M^{-1}$ (but you know the other $N^2-n$ elements ...
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0
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What are the name and inverse of an interesting integer matrix?
It is practicable to compute the matrix inverses
\begin{align*}
\begin{pmatrix}
1 & 0 & 0 \\
1 & 1 & 1 \\
1 & 2 & 2^2 \\
\end{pmatrix}^{-1}
&=\begin{pmatrix}
1 & 0 &...
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How expressive is $e^A$ in the sense of universal approximation?
For any real matrix $B\in\mathbb{R}^{n\times n},n\ge 2$ and precision $\varepsilon$, is there a real matrix $A\in\mathbb{R}^{n\times n}$ such that $\|e^A-B\|_F<\varepsilon$? ($F$ refers to ...
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Perturbation of matrices
Let $A(t)$ be a symmetric $n\times n$ matrix that continuously depend on $t\in [0,1]$. Let $\lambda_1(t)$ stand for the smallest eigenvalue for $A(t)$.
Question. Does there exist a Lebesgue measurable ...
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1
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Eigenvalues of directed Laplacian matrix $L$ and $DL$, where $D$ is a diagonal matrix with positive entries
I have a weighted Laplacian matrix $L$ of a strongly connected directed graph and a diagonal matrix $D$ with positive entries. Since the graph is directed, $L$ is non-symmetric real. Further, since ...
- 1,446
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1
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A RKHS interpretation of the Rydberg formula for hydrogen and an application for physics?
I was thinking if it is possible to define an inner product between two small physical objects with a positive definite kernel and was led to look at the Rydberg formula:
The Rydberg formula for ...
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1
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Conditions of P for existence of orthogonal matrix Q and permutation matrix U satisfying QP = PU
Question: Let $P\in \mathbb{R}^{d\times n}$ be a $d$-rank real matrix and $PP^T = c I_d$ with a certain constant $c > 0$. Under what additional conditions of $P$ does there exist an orthogonal ...
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Determining the kernel of a Vandermonde-like matrix
The kernel of a Vandermonde matrix can be determined using >this< formula.
The following type of matrix has a similar structure, and should also have a one-dimensional kernel.
$V=
\begin{...
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Special considerations when using the Woodbury matrix identity numerically
Are there any special considerations when using the Woodbury matrix identity numerically? What is the best metric for numerical stability in this case? Can anyone point me to a good reference?
The ...
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What is the quantity $\sqrt{\frac{c^2+d^2}{a^2+b^2}}$ of a matrix with determinant one?
Suppose that $A \in \mathbb R^{2 \times 2}$ has determinant one,
$$
A = \begin{pmatrix}
a & b \\
c & d
\end{pmatrix}
$$
I'm currently working on a problem where I obtained a condition on the ...
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Prime index subgroups of $\langle Q^{i}(\mathbb Z^{2}) \mid i \in \mathbb Z \rangle$ that is invariant under matrix $Q$
Let $Q $ be a matrix in $ \operatorname{GL}(2, \mathbb{Q}) $ and consider the group
$G = \langle Q^{i}(\mathbb Z^{2}) \mid i \in \mathbb Z \rangle := \langle Q^{i}(v) \mid i \in \mathbb Z, v \in \...
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Is there any generalization of the convexity of $t^{-p}$ for $p > 0$ for real symmetric positive definite matrices?
Let $p > 0$. On the positive reals, $t \mapsto t^{-p}$, is a convex function, as can be seen easily by a plot or differentiation.
However, unfortunately, unless $p \in (0, 1]$, the map $f_p(X) = X^{...
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votes
1
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Existence of finite dimensional representation of an algebra
Let $m>1$ be an integer and let $A$ be the algebra generated by the elements
$\{u^i_j,v^i_j,\bar{u}^i_j, \bar{v}^i_j| 1\leq i,j\leq m\}$ quotient over the relations
\begin{eqnarray}
u^i_j v^k_l&...
CommunityBot
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4
votes
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Conditions for distinct nonzero eigenvalues in product DAD for symmetric matrix A with repeated nonzero eigenvalues and diagonal matrix D
Let $A\in\mathbb{R}^{n\times n}$ be an real symmetric matrix with eigenvalues $\lambda_1, \lambda_2, \cdots, \lambda_n$ with some of which be nonzero and repeated, i.e., there exist $\lambda_i \ne 0$ ...
1
vote
1
answer
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square matrix depending on complex value: spectral radius continous? [closed]
Let $A(z)$ be a $n\times n$ square matrix depending on the complex value $z$ and $\lambda_z$ is its spectral radius.
Is $\lambda_z$ continous or is it possible that it can jump? Or maybe someone knows ...
- 5,632
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0
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Norm of matrix product sum
Given matrices $A_{n\times n}, B_{n\times m}, C_{m\times m}$ such that $A^iBC^{N-i}$ is matrix with all zeros except upper right element for all $i$ from $0$ to $N$, what can we say about Frobenius ...
- 1
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1
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Diameter of the unimodular group with Gauss moves
$\DeclareMathOperator\GL{GL}$Consider the unimodular group $\GL_n(\mathbb{Z})$, consisting of integral matrices $A \in \mathbb{Z}^{n \times n}$ such that that $\det(A) =\pm 1$.
It is well known that ...
- 60.4k
2
votes
0
answers
120
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Decompose a rational matrix as an integer matrix and an inverse of integer matrix
Suppose we have a non-singular rational matrix $Q$, consider the the $\mathbb{Z}$-span of the columns of $Q$ and $Q^{-1}$, denote it as $H = {\rm Span}_{\mathbb Z} \{ Q(\mathbb Z^{n}), Q^{-1}(\...
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2
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Methods to solve for a matrix whose entries satisfy certain properties
(This question is a repost of a deleted question I asked, because the previous version had several elements missing)
Setting
For fixed $N \in \mathbb{N}$, I wish to compute the entries of a matrix $...
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