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Let $V$ be a finite-dimensional vector space over a field $k$, $v_1, \dotsc v_n \in V$ a set of vectors, and $f_1, \dotsc f_n \in V^{\ast}$ a set of covectors. Up to permutation, there seem to be at l …

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 …

Well, let me say what I know so far. For monic quadratic polynomials it's necessary and sufficient that both roots be real and one be positive with absolute value at least the other. This requires no …

Suppose $P(x)$ is a monic integer polynomial with roots $r_1, ... r_n$ such that $p_k = r_1^k + ... + r_n^k$ is a non-negative integer for all positive integers $k$. Is $P(x)$ necessarily the charact …

(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 you want to compute the period of the Fibonacci sequence $\bmod p$. This reduces to examining the powers of the matrix $\left[ \begin{array}{cc} 1 & 1 \\\ 1 & 0 \end{array} \right]$ over $\mat …

I was always bothered by the definition of the cross product given in e.g. a calculus course because it's never made clear how one would go about defining the cross product in a coordinate-free manner …