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Let me elaborate a little on what Steve Huntsman is talking about. A matrix is just a list of numbers, and you're allowed to add and multiply matrices by combining those numbers in a certain way. Wh …

Recall that the adjugate $\text{adj}(A)$ of a square matrix is a matrix that satisfies $$A \cdot \text{adj}(A) = \text{adj}(A) \cdot A = \det(A).$$ Like the determinant, the adjugate is multiplicati …

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 …

The simplest case - where you only need the Banach fixed point theorem - is quite beautiful if you think about it the right way: your map lands somewhere on the land it marks, so somewhere on the map …

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 …

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 …

I've been told that in machine learning it's common to compute the singular value decomposition of matrices in order to throw out all information in the matrix except that corresponding to, say, the $ …

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 …

Perhaps it would be most appropriate to answer your question with another question: how do you distinguish a finite-dimensional vector space from an infinite-dimensional one without talking about base …

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 …

$n(n-1)...(n-(k-1))$ is the number of injective functions from a set of size $k$ to a set of size $n$. We can count these using inclusion-exclusion: first include all such functions, of which there a …

Okay, so you can get pretty close as follows: I still don't think $V^{\otimes W}$ makes sense, but riffing off of your comment, we can make sense of $(1 \oplus V)^{\otimes W}$ (where $1$ denotes the $ …

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} …

It seems to me is that there are a lot of things in mathematics that one could call hypermatrices if one were so inclined, but which people generally don't (and if they call them anything, they call t …

Not a complete answer. First, here is an alternate derivation of the result in the finite-dimensional case which might be more enlightening. If $A$ is positive self-adjoint, we can write $A = \exp(L)$ …