All Questions
Tagged with matrices linear-algebra
1,683 questions
26
votes
6
answers
14k
views
Deriving inverse of Hilbert matrix
The Hilbert matrix is the square matrix given by
$$H_{ij}=\frac{1}{i+j-1}$$
Wikipedia states that its inverse is given by
$$(H^{-1})_{ij} = (-1)^{i+j}(i+j-1) {{n+i-1}\choose{n-j}}{{n+j-1}\choose{n-...
- 1,254
10
votes
2
answers
18k
views
Fast trace of inverse of a square matrix
Which would be the most efficient way (in computational time) to compute tr(inv(H)), where H is a (dense) square matrix?
In my particular problem I also have a LU decomposition of H already available,...
- 339
5
votes
4
answers
3k
views
Parametrization of O(3)
Is there a simple way to parametrize the orthogonal group O(3) of 3 by 3 orthogonal matrices?
- 95
19
votes
1
answer
2k
views
Smallest eigenvalue of a tricky random matrix
While experimenting with positive-definite functions, I was led to the following:
Let $n$ be a positive integer, and let $x_1,\ldots,x_n$ be sampled from a zero-mean, unit variance gaussian. Consider ...
- 28.6k
16
votes
1
answer
2k
views
Playing an (invertible) matrix game with two players
Players $A$ and $B$ take an empty $n \times n$ matrix and place, one by one, an element (say, a rational number) in an unoccupied place of this matrix. Player $A$ starts. The game ends if there is no ...
- 161
5
votes
1
answer
1k
views
Techniques for lower-bounding angle between two eigenvectors of a matrix
Are there any techniques for lower-bounding the angle between eigenvectors of a matrix? Or a lower bound on the related quantity of the condition number of the matrix of eigenvectors? In particular I'...
- 303
10
votes
1
answer
1k
views
Bounds on $\|P^{k+1} - P^k\|$ for $n$ by $n$ stochastic matrix $P$ with trace $n-1$ and integer $k\gg n$
The problem:
We have a $n$-state Markov chain with arbitrary initial distribution and transition matrix $P$ that is arbitrary except that we know that $P$ has trace $n-1$. Of course $P$ is also a ...
- 303
6
votes
2
answers
2k
views
Tight bound for sum of entries of the inverse of a nonnegative matrix
While playing around with certain non-negative matrices, I got stuck at the following question.
Let $A$ be a strictly positive-definite $n \times n$ matrix ($n \ge 3$), with ones on the diagonal, and ...
- 28.6k
2
votes
1
answer
1k
views
The difficulty of generate complex Hadamard matrix
A complex $n\times n$ matrix $A=[a_{ij}]$ is called a Hadamard matrix if $A^{+}A=nI$ and $|a_{ij}|=1$ holds for all $i,j$, where $A^{+}$ denotes the conjugate transposed matrix of $A$, and a vector $...
- 1,503
8
votes
0
answers
221
views
Standard polynomials applied to matrices (bis)
The standard polynomial in $r$ non-commuting indeterminates $x_1,\ldots,x_r$ is defined by
$${\mathcal S}_r(x_1,\ldots,x_r):=\sum_{\sigma\in S_r}\epsilon(\sigma)x_{\sigma(1)}x_{\sigma(2)}\cdots x_{\...
- 52.3k
6
votes
1
answer
632
views
Norm of commutators (bis)
This question is slightly related to a popular one with the same title (see here).
Let $k$ be a field with characteristic zero. It is known (see Exercise 310) that a matrix $A\in M_n(k)$ is nilpotent ...
- 52.3k
1
vote
0
answers
221
views
Nonunique low-rank matrix completion from a few entries
Suppose we want to have a good approximation for the following NP-hard problem
$$\min_{\bf X} \operatorname{rank}({\bf X}) \text{ s.t. } \mathcal{A}({\bf X}) = {\bf b}, {\bf X} \succeq 0$$
where ${\bf ...
- 449
19
votes
1
answer
903
views
Is the norm of a $0-1$ matrix (almost) attained on a $0-1$ vector?
I'd like to state explicitly a problem which was somehow hidden in my three-week-old post:
Does there exist an absolute constant $c>0$ with the property that for any matrix $M\in{\mathcal M}_{m\...
- 23k
16
votes
4
answers
3k
views
How many minors I need to check to conclude all minors will vanish ?
Given a $m \times n$ matrix $n>m$, I was trying to check if all its $m \times m$ minor vanish.
I remember hearing that one really does not need to check all possible minors in order to conclude ...
- 1,795
1
vote
2
answers
6k
views
Square root of non-positive definite matrix
Finding square root of matrices using Cholesky decomposition is limited to positive definite matrices. Any other method to find square root of matrix which has some diagonal values approximately zero (...
- 11
0
votes
1
answer
6k
views
Finding the determinant of a matrix with LU composition
Hi Mathoverflow
I hope you bear with me that my linear algebra knowledge is a little rusty, but I have a question that might potentially very easy to answer. Nevertheless it's been bugging me for a ...
- 105
12
votes
5
answers
2k
views
Is this formulation of the Singular Value Decomposition standard?
In customary formulations of the Singular Value Decomposition or SVD that I have seen,
(e.g., Wikipedia or Gil Strang's textbooks) it is always stated in terms of writing an
$m \times n$ matrix $M$ (...
- 15.3k
5
votes
1
answer
2k
views
annihilator/common left multiple of matrix polynomials
Let $A_{n,d}$ be the space of polynomials of degree $d$ or less whose coefficients are real $n\times n$ matrices --- or, if you prefer, the space of matrices whose entries are degree-$d$ polynomials. ...
- 20.2k
6
votes
3
answers
1k
views
Pinching and positive definite matrices
A pinching over $M_n({\mathbb C})$ is an endomorphism $T$ where the $(i,j)$-entry of $T(M)$ is given either by $0$ or by $m_{ij}$, depending on the pair $(i,j)$. Let us say that a pinching is ...
- 52.3k
6
votes
2
answers
2k
views
Computation of a Drazin inverse
I need to compute the Drazin inverse $A^D$ of a singular M-matrix $A$, i.e., a matrix in the form $A=\lambda I -P$, where $P$ has nonnegative entries and $\lambda$ is the spectral radius (Perron value)...
- 20.2k
21
votes
3
answers
1k
views
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 ...
- 415
4
votes
1
answer
1k
views
Integer vectors in the kernel of an integer matrix
Let $A$ be a non-zero symmetric $n \times n$-matrix with integer entries and suppose that $\det(A) =0$.
Question: How long is the shortest non-zero integer vector in the kernel of $A$?
Example: If ...
- 25.5k
13
votes
0
answers
713
views
Regular languages of matrices and their generating functions
My question is somewhat related to this question.
Let us fix natural numbers $k$ and $C$. Let $A$ be an automaton whose alphabet consists of $k\times k$ matrices with integer coefficients of ...
- 3,897
11
votes
4
answers
5k
views
Maximum determinant of $\{0,1\}$-valued $n\times n$-matrices
What's the maximum determinant of $\{0,1\}$ matrices in $M(n,\mathbb{R})$?
If there's no exact formula what are the nearest upper and lower bounds do you know?
- 111
6
votes
1
answer
520
views
Bisymmetric Matrix, solving set of linear equations.
A bisymmetric matrix is a square matrix that is symmetric about both of its main diagonals.
If $A$ is a bisymmetric matrix and I'm interested in solving $Ax=b$.
Are there techniques used to ...
- 3,217
4
votes
0
answers
453
views
Convergence of the relaxation method for every parameter in the relevant disk
For large size matrices, the resolution of linear systems $Ax=b$ is often done iteratively. The matrix $A$ is split as $A=M-N$, with $M$ invertible, and one performs
$$x^{k+1}=M^{-1}(Nx^k+b).$$
The ...
- 52.3k
2
votes
1
answer
4k
views
Bidiagonalization and SVD of matrix
I can't find a single solid explanation of how to implement this -- whitepapers too detailed/confusing. Closest I came to an answer was this:
http://www.hep.ucl.ac.uk/~bino/libbpm/doc/pro/html/...
- 123
7
votes
3
answers
3k
views
How many commuting nilpotent matrices are there?
To be precise, fix $n$, fix a field $k$.
What is the maximal dimension of a subspace of the vector space of all $n\times n$ matrices formed by commutative nilpotent matrices? By commutative I mean ...
- 5,052
16
votes
3
answers
791
views
Random products of projections: bounds on convergence rate?
The von Neumann-Halperin [vN,H] theorem shows that iterating a fixed product of projection operators converges to the projector onto the intersection subspace of the individual projectors. A good ...
- 181
2
votes
0
answers
241
views
subspace separation and M-matrices
The separation between two square matrices $A$ and $B$, often used as a measure of the sensitivity of invariant subspace problems, is defined as
$$
\operatorname{sep}(A,B)=\min_{X\neq 0}\frac{\left\...
- 20.2k
7
votes
1
answer
372
views
Simultaneously orthogonally transform two SPD matrices to tridiagonal form?
Supposing you have two SPD matrices $A,B\in\mathbb{R}^{n\times n}$ are there any known results on the existence or non-existence of a unitary matrix $Q$ such that $Q^\top A Q=T_A$ and $Q^\top B Q=T_B$ ...
- 71
6
votes
3
answers
2k
views
Conjugate Gradient for a "slightly" singular system.
Suppose I have a symmetric $N \times N$ matrix A which has a one-dimensional Nullspace $N$. A is positive definite on $N^\bot$. In my case $N$ is the space of constant vectors (i.e. generated by ...
- 617
29
votes
3
answers
3k
views
Perron-Frobenius "inverse eigenvalue problem"
The Perron-Frobenius theorem says that the largest eigenvalue of a positive real matrix (all entries positive) is real. Moreover, that eigenvalue has a positive eigenvector, and it is the only ...
- 2,200
1
vote
1
answer
201
views
How can I characterize the type of solution vector that comes out of a matrix?
Ax = b. I need a way to analyze a square matrix A to see if its solution vector x will ...
- 133
38
votes
10
answers
18k
views
Fast matrix multiplication
Suppose we have two $n$ by $n$ matrices over particular ring. We want to multiply them as fast as possible. According to wikipedia there is an algorithm of Coppersmith and Winograd that can do it in $...
- 1,791
66
votes
2
answers
8k
views
Geometric interpretation of characteristic polynomial
The coefficients of lowest and next-highest degree of a linear operator's characteristic polynomial are its determinant and trace. These have well-known geometric interpretations. But what about its ...
- 2,071
5
votes
2
answers
2k
views
Matrices whose exponential is stochastic
The complex matrix exponential of a Hermitian matrix is unitary: $e^{-iH} = U$. Is there a name or a characterization for matrices Q whose real exponential is stochastic: $e^{-Q} = S$?
- 1,532
13
votes
2
answers
3k
views
Left and right eigenvalues
A quaternionic matrix $A$ gives rise to a
function $\mathbb{H}^n \to \mathbb{H}^n$
given by $x \mapsto A \cdot x$. This is real linear,
but not complex- or quaternionic-linear
(in general) if we ...
- 12.5k
28
votes
6
answers
5k
views
Expressing $-\operatorname{adj}(A)$ as a polynomial in $A$?
Suppose $A\in R^{n\times n}$, where $R$ is a commutative ring. Let $p_i \in R$ be the coefficients of the characteristic polynomial of $A$: $\operatorname{det}(A-xI) = p_0 + p_1x + \dots + p_n x^n$.
I ...
- 427
4
votes
1
answer
1k
views
Stability of Conjugate Gradient Method
When dealing with spd matrices of relatively low condition number, how likely is it (and is it easily provable) that the conjugate gradient method will always be able to find the solution without ...
- 221
4
votes
1
answer
496
views
Is there a standard measure for how close a matrix is to being a distance metric ?
Suppose I have a square n*n, symmetric matrix with positive elements and zero diagonal.
For this to be considered a proper distance metric between n points, the triangle inequality needs to be ...
- 711
12
votes
2
answers
8k
views
Is there a way to simplify block Cholesky decomposition if you already have decomposed the submatrices along the leading diagonal?
Let's say we have a block matrix $ M =\left( \begin{array}{ccc}
A & B\\
B^{*} & C \end{array} \right)$ where $M$ is positive definite. ($A$ and $C$ are also positive definite.)
There is a ...
2
votes
0
answers
4k
views
Eigenvalues of sum of commuting matrices [closed]
With reference to the following thread :
Eigenvalues of Matrix Sums
Answer by Jonas Meyer is as follows :
If 2 positive matrices commute, than each eigenvalue of the sum is a sum of eigenvalues of ...
- 21
1
vote
1
answer
736
views
Matrix Conjugates over Finite Fields
Thinking about Diffe-Hillman for matrices brought me to the following question.
Given $\mathbb{F}_{p^k}$ the finite field with $p^k$ elements when can we find non-trivial solutions to
$\begin{...
- 4,842
2
votes
1
answer
810
views
On matrices that almost have the same eigenvalues
Let $A$ and $B$ be two $4\times 4$ matrices. Using Newton's identities, one can prove that if
$$\det(A) = \det(B)\quad \text{and}\quad \mathrm{tr}(A^i) = \mathrm{tr}(B^i)$$ for $i=1,2,3$, then $A$ and ...
- 2,154
4
votes
1
answer
866
views
When is a triangular matrix totally unimodular?
I have a {0,1}, invertible, triangular matrix, that I would like to show is totally unimodular. Are there any known results on the total unimodularity of classes of triangular matrices?
- 1,182
15
votes
9
answers
9k
views
Exponential of large matrices
I want to make a diffusion kernel, which involves $e^{\beta A}$, where A is a large matrix (25k by 25k). It is an adjacency matrix, so it's symmetric and very sparse.
Does anyone have a ...
- 151
8
votes
3
answers
414
views
What can be said about pairs of matrices P,Q that satisfies $(P^{-1})^T \circ P = (Q^{-1})^T \circ Q$ ?
Let $P,Q$ be $n$ by $n$ invertible matrices. Suppose further that $P$ and $Q$ satisfies the following equation :
$$(P^{-1})^T \circ P = (Q^{-1})^T \circ Q$$
where $\circ$ denotes the Hadamard matrix ...
- 2,154
3
votes
2
answers
1k
views
"Main" diagonal of a matrix
Hello!
I'm in search of some (possibly statistical) measure for matrices. I want to classify a square matrix as having the largest numbers running along the main diagonal or along the anitdiagonal. ...
- 33
8
votes
4
answers
7k
views
Positive solutions of linear Diophantine equations
Let $A$ be a non-negative integer $k\times n$-matrix (i.e. each entry is non-negative and integer) with $rank(A) = k < n$. Let $b$ be a $k$-dimensional vector with positive integer entries. ...
- 351