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.
Questions tagged [matrices]
3109 questions
19
votes
3 answers
Is the semigroup of nonnegative integer matrices with determinant 1 finitely generated?
The group of $n\times n$ matrices with integer entries and determinant equal to 1, $SL(n,Z)$, is a finitely generated group (in fact, it is generated by 2 matrices). I am interested to know if the semigroup of the matrices in $SL(n,Z)$ where all the…
Hej
- 1,045
16
votes
4 answers
Diagonalizing a Complex Symmetric Matrix
For a symmetric matrix M with complex entries, I want to diagonalize it using a matrix A, such that
$AMA^T = D$, where D is a diagonal matrix with real-positive entries.
Question 1: When can this be done?
Question 2: Is $A$ unitary, i.e., is…
PoorPhysicist
- 163
15
votes
4 answers
How do eigenvectors and eigenvalues change when we remove a row/column pair of a matrix?
Let us have a symmetric matrix $C \in \mathbb{R}^{n\times n}$ having non-negative values. Suppose that we have the eigenvalue decomposition for this particular matrix such that
$$C e_i = \lambda_i e_i$$
where $e_i$ are the eigenvectors and…
İsmail Arı
- 487
15
votes
4 answers
Is there a matrix C so that the trace of C^n is dense in R?
I am looking for a matrix $C$ so that the sequence $tr(C^n)$ is dense in the set of real numbers. Equivalently (in the $2 \times 2$ case), find a complex number $z$ so that the sequence $z^n+w^n$ is dense in $\mathbb{R}$ where $w$ is the conjugate…
Hej
- 1,045
15
votes
2 answers
Comparison of the norm of a matrix and its entry-wise absolute value.
It is an easy fact that for a matrix $A \in M_n(\mathbb C)$, the matrix $A' = (|A(i,j)|)_{i,j \leq n}$ has a larger operator norm than $A$. By operator norm I mean the norm as an operator on $\ell^2_n$, or equivalently its largest singular value.
My…
Mikael de la Salle
- 9,260
13
votes
3 answers
Efficiently computing a matrix's induced p-norm
Suppose $A$ is an $m\times n$ real matrix and we need to find $\left\|A\right\|_p$ for $p \notin \{ 1, 2, \infty \}$. What is the most efficient way to compute $\left\|A\right\|_p$?
Here's one naive approach I can think of. Sample random points…
Ganesh
- 617
12
votes
2 answers
Can we add two matrices by performing an operation on their eigenvalues & eigenvectors?
Can we find out what the eigenvalues of a sum of two initial matrices are, by performing an operation on the eigenvalues & eigenvectors of the initial matrices?
Edward
- 131
11
votes
2 answers
Commuting hermitian matrices
Let $A$, $B$ be $n\times n$ hermitian matrices. Denote by $(\alpha_i)$, $(\beta_i)$, $(\gamma_i)$ the eigenvalues of $A$, $B$, and $A+B$. Assume that there exists permutations $\sigma$, $\tau \in \frak S_n$ so that
$$\gamma_i = \alpha_{\sigma(i)}…
orangeskid
- 685
11
votes
1 answer
LU factorization for $I+A$ (A skew-symmetric)
The matrix $M=I+A\in \mathbb{R}^{n\times n}$, where $I$ is the identity and $A=-A^T$ is skew-symmetric satisfies
$$
Mx\cdot x = \|x\|^2 \quad\text{for all }x\in\mathbb{R}^n.
$$
Therefore, $M=LU$ has an unique $LU$-factorization. Numerical…
Winfried
- 261
10
votes
1 answer
Arithmetic-geometric mean of positive matrices
Let $A,B$ be positive definite (Hermitian) matrices. Define the Arithmetic-geometric means of positive matrices by $A_0=A, G_0=B$, $A_{n+1}=\frac{A_n+G_n}{2}, G_{n+1}=A_n\natural G_n$, where $A_n\natural G_n$ means the geometric mean defined in…
Russel
- 223
10
votes
2 answers
spectrum of Hadamard matrices
A (±1)-matrix is a matrix whose entries are 1 and −1.
An $n \times n$ (±1)-matrix is called an Hadamard matrix if the rows are
orthogonal.
Equivalently,
An $n \times n$ (±1)-matrix $H$ is Hadamard ⇔ $H H^t = nI_n$,
where $I_n$ denotes the $n \times…
GA316
- 1,219
9
votes
3 answers
I have a very large sparse matrix, 'A', in Ax = b. What work in advance of getting 'b' can be done to reduce solving time?
This question borders between a programming and math question (more math). I have a little matrix knowledge but this is past my ability, so any help is very much appreciated.
Question
I have a very large sparse matrix (say one million rows square).…
Jordan Smith
- 203
7
votes
1 answer
How big can a commutative algebra of $n \times n$ matrices be?
What's the maximum possible dimension of a commutative subalgebra of the algebra of $n \times n$ complex matrices?
There's a theorem of Burnside saying that any commutative subalgebra of a matrix algebra can be upper triangularized. My friend…
John Baez
- 21,373
7
votes
3 answers
Is there a simple identity for the derivative of a matrix logarithm w.r.t. a real parameter?
Let $A(t)$ be an invertible square matrix that depends (differentiably) on a real parameter $t$.
It is well known that for example
$$
\frac{d}{dt} A(t)^{-1}=-A(t)^{-1}\ \dot{A}(t)\ A(t)^{-1}
$$
and
$$
\frac{d}{dt} \text{det}\ A(t)=\text{det}( A(t))\…
user19095
- 179
7
votes
1 answer
Hankel matrix commuting with a Jacobi matrix
Assume the semi-infinite Hankel matrix $H$ with entries
$$H_{i,j}=\alpha_{i+j-1}, \quad i,j\geq1,$$
where $\alpha_{n}\in\mathbb{R}$, is given. It turns out that in some special situations a semi-infinite Jacobi matrix $T$ commuting (formally) with…
Twi
- 2,188