Search Results

Here is a positive result. Every finite-dimensional algebra $A$ over a field $K$ has an intrinsic determinant, and in fact an intrinsic characteristic polynomial, which is preserved by all automorphis …

This old MO thread and its comments contains a discussion of the Zariski density proof of Cayley-Hamilton (I have also asked a separate question about the proof Victor gives in the comments here). Vic …

Okay, let me see if I've understood what all this notation means, with the help of the Wikipedia article. Let $V$ be an $n$-dimensional real inner product space and let $F : V \to \text{Cl}(V)$ be a s …

You don't say what kind of a group $G$ is but I'm going to assume for simplicity that $G$ is finite. Then, yes, it follows from Artin-Wedderburn. The point is that once we know that $\mathbb{C}[G] \co …

Maybe surprisingly, the classification appears to be smooth. I need to combine two results neither of which I understand, which is predictably fraught with danger, but here goes. The following appear …

This is the Hankel determinant associated to the sequence $m_n = \mathbb{E}(X^n) = n!$ of moments of an exponential distribution with mean $1$. Some general results can be used to show that the sequen …

Let $M_n$ be the "affine monoid scheme" of $n \times n$ matrices under multiplication (like an affine group scheme but no inverses). Claim: Every polynomial monoid homomorphism $M_n \to M_1$ is a no …

(Edit: I rewrote this answer. In the first draft I tried to take some shortcuts and found that they didn't work.) Let $G$ be a compact Lie group acting faithfully on a f.d. vector space $V$ over $\mat …

Bjorn Poonen addresses this question for commutative (associative, unital) algebras in The moduli space of commutative algebras of finite rank; asymptotically we have $$q^{\frac{2}{27} n^3 + O(n^{8/3} …

Q1: This was already given in the comments, but: a matrix $M \in GL_k(\mathbb{Z})$ of finite order $n$ must have rational normal form a block-diagonal matrix with blocks the companion matrices of cycl …

Suppose $v_j$ is an eigenbasis of $A$ with eigenvalues $\lambda_j$, so that $A v_j = \lambda_j v_j$. Then for all dual vectors $f$ we have $$\langle f, A v_j \rangle = \lambda_j \langle f, v_j \rangl …

You can use a weaker version of Jordan normal form, namely an upper triangularization. There is a very straightforward conceptual proof that every square matrix over an algebraically closed field $k$ …

The answer to Question 1 is yes, although I don't think I can extract a reasonable bound from the argument I have in mind. First observe that the question reduces to a question about largest (in absol …

An object $V$ in a symmetric monoidal category is said to be dualizable with dual $V^{\ast}$ if you can find maps $$\text{ev} : V^{\ast} \otimes V \to 1$$ and $$\text{coev} : 1 \to V \otimes V^{\as …

To my mind there are two classes of interesting categorical facts here, loosely speaking "additive" facts and "multiplicative" facts. Some additive facts: Finite-dimensional vector spaces over $k$ h …