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Results tagged with matrices
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user 102946
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.
58
votes
Accepted
Is this proof of Perron's theorem correct, and if so is it original?
There is an enormous amount of literature out there which all deals with spectral properties of positive matrices in one way or another. … which leave invariant a cone in $\mathbb{R}^n$, to eventually positive matrices, to Krein-Rutman type theorems on ordered Banach spaces whose cone has non-empty interior, and to Perron-Frobenius theory …
- 12.5k
26
votes
2
answers
1k
views
Symmetric strengthening of the Cauchy-Schwarz inequality
Or, more generally, is there a version of $(*)$ for which both sides are invariant under multiplying both $v$ and $w$ by (identical) orthogonal matrices? …
- 12.5k
14
votes
Accepted
Matrix exponential, containing a thermal state
It is actually easier to compute the first column of $e^{tM}$ for each $t \in \mathbb{R}$ rather than computing only the first column of $e^M$.
Indeed, let $u(t) = e^{tM}e_1$ for each $t \in \mathbb{ …
- 12.5k
9
votes
1
answer
509
views
Regular $p$-norm of a matrix
Let $n \in \mathbb{N}$ and $p \in [1,\infty]$ be fixed and endow $\mathbb{C}^n$ with the $p$-norm $\|\cdot\|_p$. For every matrix $A \in \mathbb{C}^{n \times n}$ we denote the operator norm of $A$ as …
- 12.5k
8
votes
3
answers
654
views
Representation theorem for matrices (reference request)
The following result for general (i.e. also non-normal) matrices is - very loosely - reminiscent of the above quoted spectral theorem:
Let $S$ denote the (Euclidean) unit sphere in $\mathbb{C}^n$ and …
- 12.5k
7
votes
0
answers
189
views
Hölder continuity of spectrum of matrices
Endow $\mathbb{C}^{d \times d}$ with the norm induced by the Euclidean norm on $\mathbb{C}^d$. It is well-known (to those who know it well, I guess) that the spectrum $\sigma(A)$ of a matrix $A \in \m …
- 12.5k
4
votes
1
answer
150
views
Mapping inclusion theorem for the numerical range
We denote the numerical range of a complex square matrix $A \in \mathbb{C}^{n\times n}$ by $W(A)$.
Let $A \in \mathbb{C}^{n\times n}$ and let $f: \mathbb{C} \to \mathbb{C}$ be, say, an entire functio …
- 12.5k
4
votes
Accepted
A matrix monotonicity question
The answer is no, in general. Here is a counterexample:
Let
\begin{align*}
X =
\begin{pmatrix}
1 & 0 \\
0 & 0
\end{pmatrix},
\quad \text{and} \quad
A =
\begin{ …
- 12.5k
4
votes
1
answer
468
views
Bicommutant theorem for commutative operator algebras
As a simple counterexample we can choose $\mathcal{A}$ to be the set of all upper triangular matrices in $\mathbb{C}^{2\times 2} = \mathcal{B}(\mathbb{C}^2)$. …
- 12.5k
4
votes
Lower eigenvectors of nonnegative matrices with zero trace
Partial answer: For the special case of self-adjoint matrices, the answer to (2) is yes. Funnily enough, this has nothing to do with the non-negativity of the matrix:
Proposition. …
- 12.5k
2
votes
Frobenius normal form of a doubly stochastic matrix
Here is a proof of the fact claimed by Perfect and Mirsky.
Proof. Let $A$ be doubly stochastic and let $B = P^TAP$ be a Frobenius normal form of $A$, which is given as in the question. Then $B$ is do …
- 12.5k
1
vote
Accepted
Spectrum of a product of a symmetric positive definite matrix and a positive definite operator
Here's a way to construct such an example:
For each integer $n \ge 1$ consider the matrices $A_n, M_n \in \mathbb{C}^{2 \times 2}$ given by
$$
M_n =
e_1 e_1^T
=
\begin{pmatrix}
1 & 0 \\ …
- 12.5k