All Questions
Tagged with matrices linear-algebra
1,683 questions
2
votes
1
answer
8k
views
Properties of eigenvalues of general nonnegative matrices
I am aware, that an answer to this question can be found via Perron-Frobenius theory or something very similar, but unfortunately I am far from being an expert in the field and I am unable to find the ...
- 83
11
votes
2
answers
9k
views
How to calculate the inverse of the sum of an identity and a Kronecker product efficiently?
I have a matrix $K$ which is the sum of a identity and a Kronecker product of two symmetric matrices as following and I want to calculate the inverse of it $K^{-1}$.
\begin{eqnarray}
K=\mathbf{I}_{mn}+...
- 607
6
votes
1
answer
333
views
construction of matrices verifying an identity
Let $A,B\in\mathbb{R}^{n\times n}$. Suppose that B is nonsingular and $AB\neq BA$. Can we always find real numbers $t_1,⋯,t_p$ such that
$$B\left(\displaystyle\prod_{i=1}^{p}(A+t_{i}B)\right)A=A\...
0
votes
0
answers
161
views
vector equation
Suppose you have an equation of the form $Hx=Ky$, where $x,y$ are vectors of length $n,m$ respectively ($m>n$) and $H,K$ are matrices of orders $n \times n,n \times m$ respectively. Is there some ...
- 6,962
10
votes
2
answers
4k
views
Perturbation theory for the generalized eigenvalue problem
Is there a standard reference for the perturbation theory of the generalized eigenvalue problem?
More specifically, I would like to get a systematic expansion for the problem
$(A_0 + \epsilon A_1)...
- 1,193
14
votes
4
answers
2k
views
Eigenvalues of a sum of Hermitian positive definite circulant matrix and a positive diagonal matrix
Suppose I have a real $n\times n$ matrix $\mathbf{C}$ that is Hermitian, positive-definite, and circulant. We know that its eigenvalues $\{\lambda_0,\ldots,\lambda_{n-1}\}$ are extraordinarily nice ...
- 907
5
votes
2
answers
780
views
A question about matrices with more details
Let $A,B\in\mathbb{R}^{n\times n}$. Suppose that $B$ is nonsingular and that there exists $m$ reals pairwise distinct $\lambda_{1},\cdots,\lambda_{r},\cdots,\lambda_{m}$ such that
$$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
0
votes
2
answers
193
views
Matrices whose range is equal to the column set [closed]
Is there such a thing?
I suppose there are two cases in which they may exist: over a finite field or infinite matrices (but let's stick to countable matrices then).
- 6,962
4
votes
3
answers
4k
views
upper bounds on a certain matrix norm
Is there some simple upper bound on $||(B^{-1}+A^{-1})^{-1}||$, where $A,B$ are $n \times n$ symmetric matrices?
- 6,962
3
votes
1
answer
447
views
Does a product of matrices have eigenvalue 1
Start by fixing invertible matrics $A_1, \ldots, A_m \in \mathbb{Z}^{n \times n}$.
For a sequence $i_1, \ldots, i_k$ we construct $A = A_{i_1} \cdots A_{i_k}$. We would like to know "Is 1 an ...
- 3,165
2
votes
1
answer
3k
views
Product of positive semidefinite and negative semidefinite matrices
Is the following statement true: If $A\succeq 0$ and $B \preceq 0$ are two $n\times n$ real-valued matrices, then $AB \preceq 0$.
If not, is it true that $\forall x\ge 0$ (i.e., all vectors in the ...
- 53
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
0
answers
1k
views
Matrix Operations Preserving Hurwitz Stability
I begin with terminology I use in the question. A real square matrix $A$ is
negative-stable if for every eigenvalue $\lambda$ of $A$, ${\mathrm{Re}}(\lambda) < 0$;
$\ast$-negative-stable if for ...
- 105
3
votes
1
answer
473
views
Relaxing commutativity. For c1,c2 find q1,q2: (1) [c1,c2]=q1c2-q2c1 (2) [q1,q2]=0, (3)...
Consider some elements c1,c2 in some ring.
Let me say that they are "relaxed commutative" if there exists two elements q1,q2,
such that the following conditions hold:
(1) $ [c_1,c_2]=c_1q_2-c_2q_1$
...
- 24.8k
2
votes
1
answer
682
views
When is a Schur complement an $M$-matrix?
Let $F=\begin{bmatrix}A & B \\\\ B^{T} & D\end{bmatrix}$ be symmetric and strictly diagonally dominant (thus an $H$-matrix). I also know that $B>0$ entrywise. What I am trying to show is ...
- 6,962
5
votes
1
answer
1k
views
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 ...
- 53
2
votes
0
answers
240
views
Copositive matrix?
I want to check under what conditions a matrix of the form $\alpha J -Q$ is copositive, where $J$ is the all-ones matrix and $Q$ is doubly nonegative (i.e. entrywise nonnegative and positive ...
- 6,962
11
votes
3
answers
1k
views
Maximum singular value of a random $\pm 1$ matrix
Define a matrix $\mathbf{A} \in \mathbb{R}^{m \times n}$ such that each element is independently and randomly chosen with probability $\frac 12$ to be either $+1$, or $-1$. Do you know any result in ...
- 199
1
vote
2
answers
574
views
matrix stability criterion
I have a $5 \times 5$ parametric nonnegative matrix and want to show that it's stable (in the sense that all eigenvalues are positive). It is not symmetric, but I do know in advance that it has 5 real ...
- 6,962
1
vote
2
answers
508
views
Sufficient conditions for inverse-positivity
I am trying to determine when a certain parametric matrix is inverse-positive (it's actually the one about which I asked in Explicit formula for Cholesky factorization in a special case, but the ...
- 6,962
3
votes
1
answer
1k
views
Explicit formula for Cholesky factorization in a special case
I have a positive definite matrix of the form $Q+sI-\alpha J$ ($s>2, 0 < \alpha <1$ and $J$ is the all-ones matrix), where $Q$ is "nice", nonnegative and known. I'd like to know if there is a ...
- 6,962
1
vote
0
answers
182
views
matrix-theoretic terminology query
Is there an accepted term for the following property?
Let $A$ be a real matrix such that all entries of the eigenvector corresponding to the least eigenvalue have the same sign.
NOTES: (1) The case ...
- 6,962
3
votes
1
answer
135
views
Mapping a subset of semi-definite matrices through arcsinus
Hi
I am meeting a problem concerning semi-definite positive matrices, and I have no clue concerning them, the classical approaches I know have not given any result, maybe people used to manipulating ...
- 1,352
5
votes
2
answers
4k
views
sparsity of QR decomposition
Hi, everyone!
I have a sparse $n \times n$ matrix $A$ with $nnz(A)$ denoting the number of non-zero entries in $A$. Now I use QR factorization to decompose $A$ into an orthogonal matrix $Q$ and ...
- 51
2
votes
1
answer
851
views
Null Space Perturbations
Hi,
I face a problem some time now (not a homework problem) and I believe it is related to matrix perturbations and how the null space behaves in these cases.
The distilled version of the ...
- 199
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, \...
4
votes
1
answer
1k
views
Generalizing inequality relating Euclidean distance & Frobenius norm to Bregman divergences such as relative entropy & von Neumann divergence
Motivation- A Special Case
Supposing $A,B\in\mathbb{S}^{m\times m}$ are symmetric positive semi-definite (SPD) matrices and $\mathbf{x}\in\mathbb{R}^m$ is a unit vector where $\|\mathbf{x}\|=1$, we ...
- 607
0
votes
1
answer
576
views
Solving Ax=b, where A is an unknown Toeplitz matrix, x and b are known.
I am trying to solve an equation of the form $Ax=b$, where $A$ is an unknown Toeplitz matrix, while $x$ and $b$ are known.
If one knows corresponding Matlab procedure, it'll be great.
- 11
4
votes
0
answers
154
views
connectivity in automata by words of length n-1
Let $A$ be a complete strongly connected automaton with $n$ states. Does always exist a word $v$ of length at most $n-1$ such that its underlying graph is connected?
That is for any pair of distinct ...
5
votes
5
answers
2k
views
median of matrices
I have $n$ positive definite Hermitian matrices $M_n$ and I want to define and compute their median.
These matrices correspond to independent estimations of a covariance matrix in the presence of ...
- 427
6
votes
1
answer
13k
views
what are the conditions for the product of 2 symmetric matrices being symmetric [closed]
In generally, the product of two symmetric matrices is not symmetric, so I am wondering under what conditions the product is symmetric.
Likewise, over complex space, what are the conditions for the ...
- 61
5
votes
1
answer
1k
views
Is Diagonalization worth to be taught? [closed]
When students come to the College (first two years of the University system in most of the developped countries) to train in mathematics, they get a linear algebra / matrix analysis course. After a ...
1
vote
3
answers
584
views
Checking for invertibility of large matrices in MAGMA
If you have a number of large matrices, and you wish to determine whether each matrix has determinant zero or not, what is the most efficient way to do this in MAGMA
(it appears that calculating the ...
- 295
5
votes
2
answers
259
views
Interlacing for "Almost Hermitian" matrices
I am wondering if there is something known about the interlacing properties of an "Almost Hermitian" matrix, in the following sense: let A be a nxn matrix so that it has a Hermitian principal minor of ...
- 6,962
1
vote
2
answers
3k
views
Low-rank factorization of SPD matrix
I have a symmetric positive definite (SPD) matrix $A$ that needs to be factorized as ${A=SS^{T}}$. However, using the Cholesky decomposition for this purpose is prohibitive in terms of computational ...
- 489
6
votes
3
answers
7k
views
Calculating the Perron-Frobenius eigenvector of a positive matrix from limited information
In the background of this question is a matrix $A$, all of whose elements are positive. The Perron-Frobenius theorem tells us that the eigenvalue with largest absolute value is real, and that there ...
- 430
6
votes
3
answers
482
views
Linear subspaces in cones over orthogonal groups
Consider the orthogonal group $G=O(n)$ as a subset of the vector space of $n\times n$ real matrices. Let $C=C(G)$ denote the Euclidean cone over $G$, i.e., the space of matrices of the form $tA, A\in ...
- 31.2k
6
votes
0
answers
465
views
Spaces of matrices with same eigenvalue/Great circles in O(n)-orbits
Let $Sym^2(V)$ be the set of symmetric matrices of a real $n$-dimensional vector space $V$. Given an element $\underline{\lambda}=[\lambda_1,\ldots \lambda_n]\in \mathbb{RP}^n$, where $\lambda_1\leq\...
- 1,452
3
votes
0
answers
220
views
Could SVD be used to optimize the partial inner-products?
Suppose a set $N$ of $n$ distinct points in $m-$dimensional space is given in $X\in\mathbb{R}^{n\times m}$. Also, suppose a subset $L\subset N$, $|L|=l<m<n$, with
$m-$dimensional coordinates in ...
- 131
2
votes
2
answers
1k
views
Positive semidefinite decomposition, Laplacian eigenvalues, and the oriented incidence matrix
Suppose $A\in\mathbb{C}^{n\times n}$ is Hermitian and positive semidefinite with some decomposition $A=BB^*$, where $B=(b_{ij})\in\mathbb{C}^{n\times m}$ (not necessarily the Cholesky decomposition). ...
- 475
6
votes
1
answer
643
views
q-analog of the matrix exponential
I am a fan of the Matrix exponential $\exp(X)$, defined for any complex matrix $X$ by
\begin{equation*}
\exp(X) := \sum_{k \ge 0} \frac{X^k}{k!}.
\end{equation*}
I have a fleeting acquaintance with ...
- 28.6k
4
votes
1
answer
386
views
Given $\mathbf{x}_i^\top A\mathbf{x}_i$ for a SPD matrix $A$ and orthonormal bases $\mathbf{x}_i$, what is the bound of its eigenvalues?
Assume that $A_{d\times d}$ is a symmetric positive semi-definite matrix, and $\{\mathbf{x}_1,\ldots,\mathbf{x}_d\}$ composes a group of orthogonal bases of $\mathbb{R}^d$ where $\mathbf{x}_i\bot\...
- 607
8
votes
0
answers
738
views
Bounding sum of first singular values squared for Kronecker sum of traceless matrices
Let $A$ and $B$ be $4\times4$ traceless matrices with Hilbert-Schmidt norms summing up to $1/4$, i.e.
$$\text{Tr}\left[ A\right]=\text{Tr}\left[ B\right] = 0,\qquad\text{Tr}\left[ A^\dagger A + B^\...
- 1,612
13
votes
1
answer
732
views
What is the "positive part" of the unit ball in $M_n(R)$ ?
In ${\bf M}_n(\mathbb R)$, let us consider the usual operator norm
$$\|A\|=\sup\frac{\|Ax\|}{\|x\|},$$
where $\|x\|$ is the Euclidian norm.
The closed unit ball $B$ is the set of contractions (in the ...
- 52.3k
2
votes
2
answers
390
views
Ax=0, estimate min(Hamming(x)) ? Equivalently: Bipartite graph. How to find (estimate) minimal number of vertices1 which are connected with EVEN number of vertices2 ? Equivalently: estimate minimal weight of error correcting code ?
Consider system of linear equations Ax=0 over $F_2$ (field with two elements {0,1}).
Where number of variables is bigger than equations - so we have many solutions $x$.
Question How to estimate ...
- 24.8k
9
votes
1
answer
1k
views
0 eigenvalue for a symmetric tridiagonal matrix
Let $T\in \mathbb{R}^{n\times n}$ be a symmetric tridiagonal matrix having the off--diagonal entries equal to -1. The diagonal entries are all positive, $a_i>0$, $i=\overline{1,n}$, and there ...
- 143
4
votes
1
answer
335
views
What is the geometry of the intersection of some cones defined by generalized inequalities?
Hello, considering that for real numbers, the intersection of intervals defined by simple inequalities has a quite simple form as
$$
\bigcap_i\{x|x\leq a_i\}=\{x|x\leq\min_i\{a_i\}\}
$$
However, what ...
- 607
3
votes
1
answer
1k
views
What is the minimum of the Frobenius norm in the intersection of positive semidefinite cones?
For scalar variables $x$, we have a simple solution for the following problem.
\begin{eqnarray}
\min_x&&\alpha(x-a)^2+\beta(x-b)^2 \\\
\mathrm{s.t. }&&x\leq a\\\
&&...
- 607
128
votes
13
answers
27k
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{...
- 7,337