Questions tagged [matrix-theory]
Matrix theory is the study of matrices as concrete objects, rather than as abstract linear operators between vector spaces (whose study belongs to linear algebra). For instance, this involves matrix factorizations and decompositions, nonnegative matrices and Perron-Frobenius theory, Schur complements, structured and special matrices, matrix functions and equations.
466 questions
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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 ...
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2
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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=\...
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2
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Decomposing a matrix into a product of sparse matrices
How to study the decomposition of a square matrix into a product of sparse matrices?
There are no restrictions on the number of matrices in the product, but the fewer the better.
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2
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Does $R$ is Dedekind-finite imply $\mathbb{M}_n(R)$ is Dedekind-finite
Following Lam's notation, a ring (with identity) $R$ is called Dedekind-finite if $ab=1\iff ba=1$ in $R$.
There are a lot of result about left invertible implies right invertible. But the results all ...
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1
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A potential new norm for matrices and Horn's inequalities
I am investigating a function defined in terms of the singular values of matrices. Initially, I simplified the problem by focusing on the eigenvalues of $2 \times 2$ Hermitian, positive-definite ...
5
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4
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Differentiability of eigenvalue and eigenvector on the non-simple case
Let $h:\mathbb{R}^n\to\mathbb{R}^m, n>1$ be a twice continuously differentiable function and $J_h:\mathbb{R}^n\to\mathbb{R}^{m\times n}$ be its jacobian matrix. Let us consider the functions $A(x):=...
- 1,638
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2
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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 ...
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Isomorphism between the hyperbolic space and the manifold of SPD matrices with constant determinant
I'm studying properties of the manifold of symmetric positive-definite (SPD) matrices and I've learnt about the following connection to the hyperbolic space [1, Section 2.2], $$\mathcal{P}(n) = \...
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2
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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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Do matrices with only elements along the main and anti-diagonals have a name?
To expand upon the title, I am wondering if there is a specific name for square matrices of the form: $$M = \begin{bmatrix} a_{11} & 0 & \cdots & 0 & \cdots & & 0 & b_{1n} ...
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2
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On the existence of integer square root of a $3 \times 3$ positive definite matrix
As far as I know, a real square matrix $M$ has a real square root if $M$ is positive semidefinite, i.e., if all eigenvalues are nonnegative. And, in fact, its square root is unique.
I have read some ...
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Stabilization of the pencil of skew symmetric matrices by the orthogonal group
During my researches I've come across the following question.
Let $A$ and $B$ be a couple of square $k\times k$ skew symmetric matrices on $\mathbb R$. Let us consider the (real) pencil generated by $...
user17697
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Best orthogonal approximation of rank 1 matrix
Let $X=\lambda_0u_0v_0^T\in\mathbb{R}^{n\times n}$ be a rank 1 matrix where $\lambda_0\in\mathbb{R}$, $u_0,v_0$ are of unit Euclidean norm. What is the solution of the following problem?
$$\hat{X}=\...
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Generalization of Jordan's Lemma $A^2=B^2=I$ can be 2-block diagonalized
One of Jordan's lemma states that if two orthogonal matrices $A,B$ are such that $A^2=B^2=I$, then they can be co-diagonalized by block of size 2.
(the proof is easy, consider $x$ an eigenvector of $A+...
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almost diagonal Positive semidefinite Matrix
Consider the set $\mathcal{D}_n$ of $n$-dimensional positive semidefinite matrices.
A matrix $M\in \mathcal{D}_n$ is called $\epsilon$ diagonal in trace distance if there is a diagonal matrix $D\in \...
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Clebsch–Gordan decomposition for $\mathrm{SU}(2)$, in indices
Let $\pi_m$, $m \geq 0$, be the unitary irreps of $\mathrm{SU}(2)$. The Clebsch–Gordan decomposition then gives that
$$ \pi_m \otimes \pi_n = \bigoplus_{k=0}^{\min(m,n)}\pi_{m+n-2k}.$$
But suppose I ...
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Can I assign the term "is eigenvector" and "is eigenmatrix" of matrix $P$ in my specific (infinite-size) case?
Remark: I asked this in MSE, the question got views and votes but seemingly no one had an answer so far.
Background: I'm rereading a couple of my exploratory (surely not research-level) math-essays ...
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Optimization of a function of a positive definite matrix and its inverse
This question is a little ill-posed, but I've been playing with some equations and am just wondering if this resembles any known problems that have been solved.
Suppose I have two real, positive ...
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Ergodic theory reference for converging sequences of matrices
I have been told that the following is a well known theorem in ergodic theory & have been given the book by Furstenberg as a reference. However, I cannot find such a statement in it. Would anyone ...
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Row-based iterative algorithms for computing the kernel of a matrix
Suppose $A$ is an $m \times n$ matrix in the form
$$A=\begin{pmatrix} — a_1 —\\ — a_2 —\\ \vdots \\ — a_m — \end{pmatrix}$$
where $a_i \in R^n$ is the $i$-th row of $A$. I know that it is possible ...
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Coloring in Combinatorial Design Generalizing Latin Square
I have a question about a combinatorial design very similar to a Latin Square, which is arising out of an open problem in graph theory. The design is an $n \times n$ matrix whose entries we want to ...
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Eigenvalues and eigenvectors of tridiagonal matrices
What can I say about the eigenvalues and eigenvectors of the tridiagonal matrix $T$ given as
$T = \begin{pmatrix}
a_1 & b_1 \\
c_1 & a_2 & b_2 \\
& c_2 & \ddots & \ddots \\
&...
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Diagonalization of real symmetric matrices with symplectic matrices
Consider the following real symmetric matrix
$M=\left[\begin{array}{ccc} A & B\\ B^T & D \end{array}\right]$
Both A and D are real symmetric $n\times n$ matrices. B is a real $n\times n$...
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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} \...
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On the linear transformation between matrix space
Suppose we have two matrix subspaces, $n\times n$ matrix subspace $S_1$ and $m\times m$ matrix subspace $S_2$. Every element of $S_1$ and $S_2$ is complex symmetric matrix.
Suppose there exists ...
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Solving $\text{trace}\left[\left(I + pY\right)^{-1} \left(I - p^{2}Y\right)\right] = 0$ for scalar $p$
For given $n\times n$ real symmetric positive definite matrices $X_{1}, X_{2}$, let $Y := X_{1}^{-1}X_{2}$, and let $I$ be the identity matrix.
I would like to solve the following equation for the ...
- 1,538
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3
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Properties of matrix $X=\left[\frac{1}{1-\bar\alpha_i \alpha_j}\right]_{ij}$
Let $\{\alpha_i\}_{i=1}^n$ be complex numbers such that $|\alpha_i|<1$, and consider the following $n\times n$ structured matrix
$$
X=\left[\frac{1}{1-\bar\alpha_i \alpha_j}\right]_{ij}.
$$
Such ...
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No arbitrary product of matrices has eigenvalue 1?
Consider the matrix $D$, adjacency matrix of an undirected graph $G$ on $n$ vertices, with the notation that $d_{i,i}=0,\forall i$.
The matrices $A_i$ are constructed from Identity matrices, $I_{n*n}$...
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Is Ryser's conjecture on permanent minimizers still open?
Let $A(k,n)$ be the set of $\{0,1\}$ matrices of order $n$ with all their line sums equal to $k$.
Conjecture number 5 on the list from Minc's book, attributed to Ryser, says that if $A(k,n)$ ...
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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 ...
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Probability of a submatrix to be full rank in a N x N Random Matrix of rank m.
Consider a random matrix $\mathbf{A} \in \mathbb{C}^{N \times N}$ of rank $m$ with $m < N$ that follows the Wishart distribution ( http://en.wikipedia.org/wiki/Wishart_distribution ).
I have a ...
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Frobenius normal form of a doubly stochastic matrix
If $A \in M_n(\mathbb{C})$, then $A$ is called reducible if there is a permuation matrix $P$ such that
$$
P^\top A P =
\begin{bmatrix}
A_{11} & A_{12} \\
0 & A_{22}
\end{bmatrix}, $$
in ...
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2
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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 $\lambda_{2}-\lambda_{1}$. (...
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Total positivity tests: optimal in the number of minors vs. the computational cost
A real matrix is called totally positive if all of its minors are positive. Of course, to check that a given matrix is totally positive it is not necessary to check all the minors: for example, it ...
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What is this matrix decomposition called and does it exist always?
Given a rank $2r$ matrix $M\in\Bbb Q^{n\times n}$ can we find two matrices $M_+\in\Bbb Q_{\geq0}^{n\times n}$ and $M_-\in\Bbb Q_{\geq0}^{n\times n}$ each of rank $r$ such that $M=M_+-M_-$ holds?
...
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A variant of Specht's Theorem using sum of elements (rather than trace) of complex matrices?
Let us first recall Specht's Theorem. Denote by $\text{Mat}_{\mathbb{C}}(n)$ the set of all $n\times n$ matrices over the complex field $\mathbb{C}$. Let $A$ be a matrix in $\text{Mat}_{\mathbb{C}}(n)$...
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A Handbook of Matrix Factorizations
I am looking for a good collection of facts regarding the various types of matrix factorizations, something like a "Handbook of Matrix Factorizations" or a very-thorough review paper. I am hoping for ...
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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 ...
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Why do these polynomials split almost in the middle?
Start with a palindromic sequence of integers $(a_0, a_1, \ldots, a_{n+1})$, i.e. $a_j=a_{n+1-j}$, and put $a_j:=0$ for $j<0$ and $j>n+1$. You may readily guess that the choice of the binomial ...
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1
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Singular value decomposition of truncated discrete Fourier transform matrix
Let $\mathbf{F}$ be a discrete Fourier transform (DFT) matrix such that
\begin{align}
F_{m,n}=e^{-j2\pi(m-1)(n-1)/N},\quad m,n=1,\ldots,N.
\end{align}
What we can say about the singular value ...
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A property of positive matrices
Let $\mathbb{S}^{dn} \ni X \succeq 0$ with $d,n \in \mathbb{N}$, where $X \succeq 0$ indicates that $X$ is positive semidefinite. Now partition $X$ into the block form
\begin{gather}
\begin{pmatrix}
...
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Positive system of algebraic integers
Let $\mathbb{A}$ be the ring of algebraic integers. Consider a sequence $(d_i)_{i \in I}$, with $I$ a finite set and $d_i \in \mathbb{A} \cap \mathbb{R}_{\ge 1}$, such that $$d_i d_j = \sum_{k \in I} ...
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Is it hard to decide whether a matrix is a square of another matrix?
According to the well-know quadratic residue (QR) theory over integers, we know that it is hard to decide whether a given integer $m\in\mathbb Z_N$ is a quadratic residue (i.e., a square of another ...
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1
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How to compute an indefinite generalisation of QR decomposition
Given an arbitrary complex matrix $M$ and real, diagonal but possibly indefinite matrix $\Delta$, the problem is to solve the following system of equations:
$$\begin{aligned}
M^*\Delta M &= LD^2L^*...
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Inverse direction of Hodge index theorem
The Hodge index theorem states that the intersection matrix associated to curves on a smooth algebraic surface has a specified signature---namely, if the intersection matrix has size $n \times n$ then ...
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A question on products of linear combinations of complex matrices
Suppose that $A_1, \dots , A_n, B_1, \dots , B_n \in \Bbb C^{d \times d}$ and that, for every $x \in \mathbb{C}^n$, the following holds
$$\left( \sum_i x_i A_i \right) \left( \sum_i x_i A_i \right)^{...
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Is every stable matrix orthogonally similar to a $D$-skew-symmetric matrix?
Let $\mathrm{diag}(A)$ denote the diagonal matrix with diagonal entries of $A\in\mathbb{R}^{n\times n}$ and let $\succeq$ denote the standard partial order in the cone of (symmetric) positive definite ...
- 2,712
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Generalizing Autonne-Takagi factorization
Autonne-Takagi factorization (Léon Autonne (1915) and Teiji Takagi (1925)) says that:
A complex symmetric matrix can be 'diagonalized' using a unitary matrix: If $A$ is a rank-$n$ complex symmetric ...
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435
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An upper bound on the Jordan condition number of a matrix
The Jordan condition number of a matrix $A$ is defined to be $\min_{V}\kappa(V)$, where $V$ ranges over complex matrices that satisfy $A = VJV^{-1}$ for $J$ being the unique Jordan normal form matrix ...
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0
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Inducing surjections on $GL_n(-)$?
Suppose $A,\,B$ are (possibly noncommutative) rings, and $GL_n(-)$ is the group of invertible $n\times n$ matrices over a given ring. Suppose $f:A\to B$ is surjective, does it necessarily follow that $...
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