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Results tagged with matrices
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user 4213
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.
1
vote
Accepted
Eigenvalues of sum of two anti-commuting matrices
Suppose for simplicity's sak that $A$ and $B$ are diagonalizable over $\mathbb{R}$
and are non-singular.
Let $V_a$ be the $a$-eigenspace of $A$. Then by anti-commutativity,
we find $BV_a\subseteq V_{ …
- 20.8k
1
vote
Accepted
Operation of GL_n(Z/bZ)
Any transformation
$$(v_1,\ldots,v_n)\mapsto (v_1,\ldots,v_{j-1},v_j+av_k,v_{j+1},\ldots,v_n)$$
for $j\ne k$ is achievable by means of some such matrix. It suffices to reduce
an admissible vector to $ …
- 20.8k
4
votes
Accepted
Invariant quadratic forms of irreducible representations
There are certainly examples over $k=\mathbb{Q}$ where $\dim T\ge2$.
Let's take the cyclic group $G$ of order $5$ and the representation
space
$$V=\{(a_0,\ldots,a_4)\in\mathbb{Q}^5:a_0+\cdots +a_4=0\} …
- 20.8k
3
votes
Matrix Conjugates over Finite Fields
This occurs if and only if the matrices $Q^r$ and $Q^s$ are conjugate.
This is the case if and only if these matrices are conjugate over the
algebraic closure of $\mathbb{F}_p$. …
- 20.8k
3
votes
Binary matrices with constant row and column sums
See this paper by Canfield and McKay. As the title suggests it focuses
on the asymptotic enumeration, but it has lots of useful references.
Added A simple arithmetic construction that realizes all pos …
- 20.8k
12
votes
Maximum determinant of $\{0,1\}$-valued $n\times n$-matrices
I shouldn't expect there to be exact results; compare the similar problem
with matrices with entries $\pm1$. … The determination for which $n$ Hadamard matrices
exist still resists proof. …
- 20.8k
12
votes
Accepted