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Pseudospectral Mirage of Matrices We usually judge a matrix by its eigenvalues, but sometimes the eigenvalues are only the bait. The real monster is hiding in the pseudospectrum. Most of the time, a matrix is shown as a few points in the complex plane called eigenvalues. But for some...

29,171 görüntüleme • 19 gün önce •via X (Twitter)

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The Trap in Every Mathematics Lecture If you’ve taken a lot of math courses, you start to recognize a pattern. There’s a moment where the lecturer is warming up with the obvious stuff...add matrices entrywise, scale by α, do the row-column product...and you’re thinking, alright… where is this going? Then you relax. You stop resisting. And right there, they slip in one line that changes how you see the whole subject. When Benedict Gross says "matrices represent linear operators,"he’s telling you to stop treating a matrix as a rectangle of numbers and start treating it as an action. A linear operator is a function T: Rⁿ → Rⁿ that respects two rules: T(u+v)=T(u)+T(v) and T(αu)=αT(u). Once you pick a basis, T is completely determined by where it sends the basis vectors e₁,…,eₙ. Put T(e₁),…,T(eₙ) into columns and you get a matrix A. That is what "A represents T" means...A is the coordinate portrait of the transformation. Now the punchline that makes matrix multiplication feel inevitable. If B represents S and A represents T, then doing S first and then T is the composition T∘S. In coordinates that becomes A(Bx)=(AB)x. So multiplying matrices is really composing transformations. That’s why multiplication is usually not commutative: T∘S is generally not the same transformation as S∘T, and the matrices inherit that noncommutativity. This explains half of Linear Algebra because it tells you what the course is really about...functions that move vectors around, not grids of numbers. A matrix is just the written form of that function once you choose coordinates. Then the rules stop feeling random Multiplying matrices means doing one move and then another, an inverse means you can undo the move, eigenvectors are directions that don’t get turned, and changing basis is just describing the same move in a different language. That one idea makes a lot of linear algebra click. #LinearAlgebra #Matrices #GroupTheory #GLn #MathLectures #Mathematics

Mathelirium

66,204 görüntüleme • 6 ay önce

The Trap in Every Mathematics Lecture If you’ve taken enough math courses, you start noticing the same little move. The lecturer warms up with the obvious stuff, add matrices entrywise, scale by α, do the row-column product, and you’re thinking alright, where is this going. Then you relax. You stop resisting. And right there, they drop one line that quietly rewires the whole subject. When Benedict Gross says matrices represent linear operators, he’s telling you to stop treating a matrix as a rectangle of numbers and start treating it as an action. A linear operator is a function T: ℝⁿ → ℝⁿ that respects two rules: T(u+v) = T(u) + T(v) T(αu) = αT(u) Once you pick a basis, T is completely determined by where it sends the basis vectors e₁,…,eₙ. Put T(e₁),…,T(eₙ) into columns and you get a matrix A. That is what A represents T means. A is the coordinate portrait of the transformation. Now the punchline that makes matrix multiplication feel inevitable. If B represents S and A represents T, then doing S first and then T is the composition T∘S. In coordinates that becomes A(Bx) = (AB)x. So multiplying matrices is really composing transformations. That’s why multiplication is usually not commutative. T∘S is generally not the same transformation as S∘T, and the matrices inherit that noncommutativity. This explains half of linear algebra because it tells you what the course is really about: functions that move vectors around, not grids of numbers. A matrix is just the written form of that function once you choose coordinates. After that, the rules stop feeling random. Multiplying matrices means doing one move and then another. An inverse means you can undo the move. Eigenvectors are directions that don’t get turned. Changing basis is just describing the same move in a different language. One idea, and a lot of linear algebra suddenly clicks. #LinearAlgebra #Matrices #LinearMaps #Eigenvectors #ChangeOfBasis #Mathematics

Mathelirium

133,454 görüntüleme • 5 ay önce

THE MATRIX SPOON BOY'S MESSAGE Many have seen the Matrix but it seems fewer grasp just how powerful the 'spoon boy's' message was when Neo first visits the oracle... The boy says to Neo: "Do not try and bend the spoon, that's impossible. Instead, only try to realize the truth...there is no spoon. Then you'll see that it is not the spoon that bends, it is only yourself." What he's saying is there is nothing that is NOT yourself, BECAUSE 'Neo' was not just an entity as the body, 'Neo' was the entire Matrix becoming self aware to re-write and change itself - this is why in the final movie there is rain and in the very end there is sunshine. Anyone who knows the Matrix knows EVERYTHING which appears is really shaped through Matrix code which means 'Neo' (who could visibly see this code) was all of the characters and places, not just the one we saw in the flesh as 'Neo'. Why do you think in the end Mr Smith asks Neo if he realises its impossible to win, that he simply can't.... And the reason nothing can be 'won' is they are not separate and Neo began realising himself as 'The Matrix' itself rather than just a body which is why he no longer fights Smith and allows him to enter his body.... The body was not 'Neo'.... 'Neo' becoming the chosen one is acknowledging the self as the entire Matrix, as one self and not two...(Mr. Anderson and Smith) There is no body which is separate from earth, only MIND has made this concept. Earth and body are ONE beyond what we call them... We are not Mr Anderson nor Neo, we are the entire Matrix writing itself to experience itself. Realise everything in the form which changes is only yourself because the self is not just limited to this body... Each and everyone of you are this 'CHOSEN ONE'. But what did Morpheus say to Neo... "DON'T THINK YOU ARE, KNOW YOU ARE"..... Until The One believes... They remain as just a character feeling stuck inside of something or somewhere when there is no where or place separate from your true self - which is why in the beginning it says: "The Matrix Has You..." In the beginning he existed as an illusory identity of Mr Anderson, the Matrix had him but he did not have the Matrix. He was not aware of the self beyond identity and form so through meeting Morpheus, "he died" and realised himself as the "chosen one" who can directly change and save his experience. What this means is ANYONE of these characters could of became 'Neo' because Neo is a realisation of the true self similar to "Christ Consciousness" as we know it... We are ALL this chosen one but until this is realised, we will never find it but seek it.... You do not become it, you ARE already IT... Any process is like running past the finish line after you've won a race, if you continue running, when do you acknowledge you've already finished... ✨🙌🏾💫 © Awakened Gnostic

🧬Maxpein🧬

129,113 görüntüleme • 1 yıl önce

THE MATRIX SPOON BOY'S MESSAGE Many have seen the Matrix but it seems fewer grasp just how powerful the 'spoon boy's' message was when Neo first visits the oracle... The boy says to Neo: "Do not try and bend the spoon, that's impossible. Instead, only try to realize the truth...there is no spoon. Then you'll see that it is not the spoon that bends, it is only yourself." What he's saying is there is nothing that is NOT yourself, BECAUSE 'Neo' was not just an entity as the body, 'Neo' was the entire Matrix becoming self aware to re-write and change itself - this is why in the final movie there is rain and in the very end there is sunshine. Anyone who knows the Matrix knows EVERYTHING which appears is really shaped through Matrix code which means 'Neo' (who could visibly see this code) was all of the characters and places, not just the one we saw in the flesh as 'Neo'. Why do you think in the end Mr Smith asks Neo if he realises its impossible to win, that he simply can't.... And the reason nothing can be 'won' is they are not separate and Neo began realising himself as 'The Matrix' itself rather than just a body which is why he no longer fights Smith and allows him to enter his body.... The body was not 'Neo'.... 'Neo' becoming the chosen one is acknowledging the self as the entire Matrix, as one self and not two...(Mr. Anderson and Smith) There is no body which is separate from earth, only MIND has made this concept. Earth and body are ONE beyond what we call them... We are not Mr Anderson nor Neo, we are the entire Matrix writing itself to experience itself. Realise everything in the form which changes is only yourself because the self is not just limited to this body... Each and everyone of you are this 'CHOSEN ONE'. But what did Morpheus say to Neo... "DON'T THINK YOU ARE, KNOW YOU ARE"..... Until The One believes... They remain as just a character feeling stuck inside of something or somewhere when there is no where or place separate from your true self - which is why in the beginning it says: "The Matrix Has You..." In the beginning he existed as an illusory identity of Mr Anderson, the Matrix had him but he did not have the Matrix. He was not aware of the self beyond identity and form so through meeting Morpheus, "he died" and realised himself as the "chosen one" who can directly change and save his experience. What this means is ANYONE of these characters could of became 'Neo' because Neo is a realisation of the true self similar to "Christ Consciousness" as we know it... We are ALL this chosen one but until this is realised, we will never find it but seek it.... You do not become it, you ARE already IT... Any process is like running past the finish line after you've won a race, if you continue running, when do you acknowledge you've already finished... ✨🙌🏾💫 © Awakened Gnostic

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59,563 görüntüleme • 8 ay önce