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Results tagged with matrices
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user 2757
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.
46
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Determinant of a determinant
Consider an $mn \times mn$ matrix over a commutative ring $A$, divided into $n \times n$ blocks that commute pairwise. One can pretend that each of the $m^2$ blocks is a number and apply the $m \time …
- 23.8k
30
votes
Integer matrices with no integer eigenvalues
More precisely, there is no algorithm that takes as input two $n \times n$ integer matrices and decides whether the semigroup they generate contains a matrix all of whose eigenvalues are integers. … An element of the semigroup generated by these two $np \times np$ matrices has the form $M \otimes C^m$ for some $M$ in the semigroup generated by $A$ and $B$ and some $m \ge 1$. …
- 23.8k
22
votes
Accepted
Is there a matrix C so that the trace of C^n is dense in R?
The answer is yes, even in the $2 \times 2$ case. Let $q_1,q_2,\ldots$ be an enumeration of the rational numbers. Let $Q_j$ be the closed interval $[q_j-1/j,q_j+1/j]$. Let $I_0=[0,2\pi]$. Let $z=2 …
- 23.8k
49
votes
Expected determinant of a random NxN matrix
If $N \ge 2$, then the expected value is $0$ since interchanging two rows preserves the distribution but negates the determinant.
- 23.8k