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Results tagged with matrix-analysis
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user 102946
The study of the properties of real and complex matrices that are more close to analysis and operator theory. For instance: the properties of positive definite matrices, matrix inequalities, perturbation analysis, matrix functions, inequalities between eigenvectors and singular values, majorization.
8
votes
3
answers
642
views
Commutant of the conjugations by unitary matrices
Let $\mathcal{L}(\mathbb{C}^{n \times n})$ denote the algebra of all linear mappings from $\mathbb{C}^{n \times n}$ to $\mathbb{C}^{n \times n}$ and let $\mathcal{C} \subseteq \mathcal{L}(\mathbb{C}^{ …
- 201
8
votes
3
answers
654
views
Representation theorem for matrices (reference request)
Motivation. If $A \in \mathbb{C}^{n \times n}$ is self-adjoint (or, more generally, normal), then we all know that
$$
A = \sum_{k=1}^n \lambda_k \, h_k \otimes h_k,
$$
where $\lambda_1,\dots,\lambda_n …
- 52.3k
7
votes
Accepted
On approximate simultaneous diagonalization
The answer is no in general.
For a $2\times 2$-counterexample, let $A = 0$, let $B$ be the diagonal matrix with diagonal entries $1$ and $0$ (i.e. $B$ is the projection onto the first component), cho …
- 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 …
- 10.1k
58
votes
Accepted
Is this proof of Perron's theorem correct, and if so is it original?
(1) Correctness: I read all arguments in detail and couldn't find anything wrong with them. Of course, this doesn't mean too much...
(2) Orginality: I think in a topic which has such an extensive his …
- 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