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Results tagged with matrices
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user 169474
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.
6
votes
Accepted
Existence of weird complex norms
Consider the norm $\Vert\cdot\Vert$ on $\mathbb{C}^2$ defined by $\Vert z \Vert^2 := |z_1|^2+|z_1+z_2|^2$. Then
$$\left\Vert \left(\begin{array}{c}1 \\ -1 \end{array}\right) \right\Vert^2 = 1,$$
$$\le …
- 3,808
2
votes
Wold decomposition of toral endomorphisms
The Hilbert space $H := L^2(\mathbb{T}^d,dx)$ can be canonically identified with the subspace of all (classes of) $\mathbb{Z}^d$-periodic locally square-integrable functions on $\mathbb{R}^d$. Then, f …
- 3,808
-1
votes
Submatrices of matrices in $\mathrm{SL}(4, \mathbb{Z})$ with all eigenvalues equal to $1$
It is not a solution since $A$ is not in $GL_2(\mathbb{Z})$, I will try to correct it later.
Let $a,b,c,d$ be in $\mathbb{Z}$ such that $ad-bc=1$.
Set
$$P = \left(\begin{array}{cc}
aI_2 & bI_2 \\
cI_ …
- 3,808