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Results tagged with matrices
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user 160416
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.
2
votes
Accepted
Upper bound of rank of a matrix
Both matrices are invertible since $B$ is a permutation matrix and $C$ is upper-triangular (with respect to any linear extension of the inclusion ordering on $2^X$) with $1$'s on the diagonal. …
- 3,446
8
votes
Accepted
On the coefficients that appear in finite groups of matrices with integer entries
We consider the conjugate by $B$ of the group of diagonal $\pm 1$ matrices, of size $2^n$.
Let $X$ be a diagonal matrix with $X_{i,i}=\varepsilon_i \in \{\pm 1\}$ and suppose $\varepsilon_n=1$. … }^{n-1} \varepsilon_i x_i$$
If we take, for example, $x_i=2^i$, then these sums are all different for the $2^{n-1}$ possible choices of $X$, so at least $2^{n-1}$ different coefficients appear in the matrices …
- 3,446
4
votes
Accepted
Similarity of two matrices
I will show that it is not possible for $\phi=\pi/2$, so it is certainly not for general $\phi$. (actually, I don't think that it is possible for any single $\phi$ except $0$ and $\pi$, by an analogou …
- 3,446
5
votes
Accepted
Eigenvalues invariant under 90° rotation
Assume $N$ is even (this is false when N is odd).
Let $X=2B, Y=A+A^T$.
Let
$$P = \begin{bmatrix}
1 & 1 & 1 & \cdots & 1 \\
1 & \zeta & \zeta^2 & \cdots & \zeta^{N-1} \\
1 & \zeta^2 & \ze …
- 3,446
20
votes
Accepted
Eigenvalue pattern
The explanation is pretty simple with a suitable change of basis.
Letting
$$B =
\begin{pmatrix}
1 & 0 & 1 & 0 \\
i & 0 & -i & 0 \\
0 & 1 & 0 & 1 \\
0 & i & 0 & -i
\end{pmatrix}$$
we have
$$B^{-1}M …
- 3,446
11
votes
1
answer
324
views
$2$-adic valuation of Schur $P$-functions in the power-sum basis
For a partition $\lambda$, let $P_\lambda$ be the Schur $P$-functions (case $t=-1$ of Hall-Littlewood symmetric functions) and let $p_\lambda=p_{\lambda_1}p_{\lambda_1}\cdots p_{\lambda_k}$ be the pow …
- 3,446