Here's a question: let \(M\) be a \(0\times 0\) matrix with entries in the field \(\mathbb{F}\). What is \(\det(M)\)?

#Mathematics #Determinant #LinearAlgebra #Matrix

https://www.patreon.com/posts/rule-of-sarrus-129154966?utm_medium=clipboard_copy&utm_source=copyLink&utm_campaign=postshare_creator&utm_content=join_link

#mathematics #visualization #determinant

Murine #betacoronavirus #spike protein: A major #determinant of #neuropathogenic properties, https://etidiohnew.blogspot.com/2025/03/murine-betacoronavirus-spike-protein.html

impossible

https://kbin.melroy.org/media/6b/b0/6bb06f22b113fc72388d256857923fdae00938579bf4b73f112493e476876d8a.png

What Numbers Do You Get by Iteratively Scaling a Matrix?
https://www.youtube.com/watch?v=-uIwboK4nwE
#maths #mathematics #Sinkhorn #matrix #determinant #scaling #math

For all Transf ∈ ℝn×n, the matrix is invertible if and only if rank(Transf) = n

The #determinant exists if and only if the transformation matrix is square.
The determinant in a linear transformation is the (signed) area of the image of the fundamental basis formed by the unit square.

#algebra #matrices #tutorial #determinants #singularity #math #maths #mathematics #mathStodon #ML #machineLearning #systems

Yeah o/ Mon premier article en anglais, co-écrit avec un collègue, est paru aujourd'hui dans le "Journal of French Language Studies" ! Je suis joie :) Ça parle de corpus, de déterminants, d'histoire du français et y'a, genre whatmille tableaux. Yeah :D

#Linguistics #Linguistique #Grammaire #Déterminant #LangueFrançaise

https://www.cambridge.org/core/journals/journal-of-french-language-studies/article/evolution-of-bare-nouns-in-the-history-of-french-the-view-from-calibrated-corpora/A809D923F7CD0854DBB9833DFAB4C428

Why is the determinant of the matrix \( \begin{pmatrix} a_1 & b_1 \\ a_2 & b_2 \end{pmatrix} \) equal to \( a_1 b_2 - a_2 b_1 \)?

I have found a geometrical interpretation (https://functor.network/user/414/entry/299) and with it also started a blog.

#WordsAndSomeFormulas #Mathematics #MathEdu #Determinant #Geometry

DETERMINANTS AND THE BAREISS ALGORITHM

If you have to calculate determinants, and especially if you have to program an algorithm, investigate the Bareiss algorithm. It's remarkably fast; it limits the divisions so that it doesn't introduce needless rounding errors; and if your matrix elements are all integers, Bareiss is guaranteed to give you an integer result.

I've worked out a way to do Bareiss on pen and paper; here's a link to a PDF showing the technique:

http://www.paprikash.com/lou/bareiss.pdf

#determinant #bareiss

Hm... I think I will do a little visual intuition #LiaScript doc about the determinant and how it corresponds to area/volume 🤔 I feel that many people are very confused by the #determinant when they encounter it in a linear algebra #math class. At least in mine, it was this weird abstract function (granted, this way of going about things also has its positives), not like a geometric thing. Then you see it popping up in like multivariate integrals with volume elements and its like... huh?

I'm reading Artin's #GaloisTheory and I'm wondering if the #math exposition of the #matrix #determinant relying on homogeneity properties is related to the #BrunnMinkowski #inequality

I will check my handwritten notes when I'll be in Poland end of July but I remember that homogeneity played a role there.

This rambling is related to my struggle to understand the #BrascampLieb and #rearrangementInequality. Any hints much appreciated 🥹😅🙈😊

Screenshot Saturday Mondays: dual-wielding katanas and duel-wielding fruit - https://www.rockpapershotgun.com/screenshot-saturday-mondays-swish-movement-and-marble-runs #ScreenshotSaturdayMondays #ScreenshotSaturday #SilentSanticado #WhimsicalHeroes #Indiescovery #Withersworn #Determinant #RetroSpace #MortalSin #AmidEvil #Enchain #Indie

In the Laplace or cofactor expansion of the determinant of an \(n\times n\) matrix, the number of operations:
\[\text{Addition: }\mathcal{A}(n)=n!-1\]
\[\text{Multiplication: }\displaystyle\mathcal{M}(n)=n!\sum_{k=1}^{n-1}\dfrac{1}{k!}=\lfloor(e-1)\cdot n!\rfloor-1\]
\[\text{Both combined: }\displaystyle\mathcal{T}(n)=n!\sum_{k=0}^{n-1}\dfrac{1}{k!}-1=\lfloor e\cdot n!\rfloor-2\]
#LaplaceExpansion #Determinant #LinearAlgebra #Cofactors #CofactorExpansion #Matrix #Operations #Algorithm #Minors

#NS2 is a key #determinant of compatibility in #reassortant #avian #influenza virus with heterologous #H7N9-derived NS segment, https://doi.org/10.1016/j.virusres.2022.199028

#determinant #linearalgebra

https://en.wikipedia.org/wiki/Quasideterminant
Wikipedia: Quasi determinant

https://arxiv.org/abs/math/0208146
Quasideterminants
I. Gelfand, S. Gelfand, V. Retakh, R. Wilson

#determinant #linearalgebra

私の数学の専門は、パンルヴェ系(古典的なパルンルヴェ方程式の大幅な一般化)のτ函数の正準量子化なのですが、最初は非可換行列式を使ってそれを遂行し切ろうとして失敗しました。

量子化されていても、τ函数の話なので、非可換な場合のGauss分解とも当然関係するのですが、私が示したかったことは量子化されたWeyl群双有理作用で生成されたτ函数が従属変数の(非可換)多項式になること(正則性)だったのですが、非可換行列式は行列の成分の非可換多項式にならず、(非可換)有理函数になるので、非可換行列式を使ってτ函数の正則性を示すことは困難で私には無理でした。

(量子化する前の対応する結果の証明は本質的に行列式が多項式になることに帰着している。非可換だとその方法は使えない。)

結局、Kac-Moody代数のBGGでのtranslation functorsのよい性質に量子τ函数の正則性が帰着することによって証明したいことを示せました。

圏の構造に関する非自明な結果は恐ろしく非自明な公式の源泉の一つになっています。

https://mathtod.online/@mathmathniconico/823004

https://mathtod.online/@mathmathniconico/823033

#determinant #linearalgebra

非可換行列式の一種に quasi-determinant というのがあります。それはちょうど表示名さんの方法による非可換な場合の行列式の話になっています。

正方行列を (対角成分が1の下三角行列)×(対角行列)×(対角成分が1の三角行列)の形に表わすことはGauss分解などと呼ばれることがあるようですが、Gauss分解は行列式の言葉で書けます。

そして、Gauss分解の真ん中の対角行列の成分は別名「τ函数」と呼ばれています。

佐藤幹夫さんによる所謂「ソリトン方程式の佐藤理論」のτ函数は行列式で書けるのですが、行列式で書ける理由はまさにGauss分解そのものなのです。