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Results tagged with matrices
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user 1306
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.
13
votes
Accepted
A strange matrix equality
.
$$
Now, for matrices $X$ of size~$2$, we have $X^2=tr(X)X-det(X)I$ (a particular case of Cayley--Hamilton), so
$$
tr(B)[tr(A)A-det(A)I,B]=tr(A)tr(B)[A,B]=tr(A)[A,tr(B)B-det(B)I],
$$
since $I$ commutes … This might be the most economic proof of your identity; moreover, it is known (Procesi, Razmyslov) that every identity with traces for $n\times n$-matrices does follow from the Cayley--Hamilton identity …
- 16.9k
6
votes
Accepted
Cayley-Hamilton over super rings
The case when $R$ is purely even but the module has a $Z/2$-grading was studied before, see for example
"On the Cayley-Hamilton equation in the supercase" by Issai Kantor and Ivan Trishin, Comm. in A …
- 16.9k
6
votes
Accepted
Set of integer matrices $A$ such that $(A^k)_{k\in\mathbb{N}}$ is eventually periodic
Of course. The eigenvalues of this matrix (over $\mathbb{C}$) may only be zeros and roots of unity (whose minimal polynomial is of degree at most $n$, as they are roots of the characteristic polynomia …
- 16.9k
3
votes
Algebraic axiomatization for AB+BA^T operation on matrices
Including the operation $A\mapsto A^T$ can be viewed, in the language of operads, in many different closely related ways: via adjoining a new unary operation $J$ that satisfies $J^2=\operatorname{id}$ …
- 16.9k