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The single most undervalued fact of linear algebra: matrices are graphs, and graphs are matrices. Encoding matrices as graphs is a cheat code, making complex behavior simple to study. Let me show you how!

Tivadar Danka

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158,849 просмотров • 7 месяцев назад •via X (Twitter)

Образование Наука и технологии

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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 matrices, especially non-normal matrices, those few points do not tell the whole story. In this scene, the complex plane becomes a moving landscape. The bright anchors are the true eigenvalues. Around them, huge pseudospectral continents grow, split, and collapse as the matrix changes. The moving threads show where the matrix is most sensitive. In those regions, even a tiny change can create a huge response. This is the hidden instability of a matrix made visible. Algebra becomes a pressure map and the pseudospectrum becomes terrain. #Mathematics #LinearAlgebra #SpectralTheory #NumericalAnalysis #MatrixTheory #MathArt #MathematicalArt #STEM #Art

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 matrices, especially non-normal matrices, those few points do not tell the whole story. In this scene, the complex plane becomes a moving landscape. The bright anchors are the true eigenvalues. Around them, huge pseudospectral continents grow, split, and collapse as the matrix changes. The moving threads show where the matrix is most sensitive. In those regions, even a tiny change can create a huge response. This is the hidden instability of a matrix made visible. Algebra becomes a pressure map and the pseudospectrum becomes terrain. #Mathematics #LinearAlgebra #SpectralTheory #NumericalAnalysis #MatrixTheory #MathArt #MathematicalArt #STEM #Art

Mathelirium

29,171 просмотров • 1 месяц назад

New package + paper drop 📄 - Introducing KGGen – a simple library to transform unstructured text into knowledge graphs. Text is abundant, but good knowledge graphs are scarce. Feed it raw text, and KGGen generates a structured network of entities and relationships. (1/7)

New package + paper drop 📄 - Introducing KGGen – a simple library to transform unstructured text into knowledge graphs. Text is abundant, but good knowledge graphs are scarce. Feed it raw text, and KGGen generates a structured network of entities and relationships. (1/7)

Belinda

11,111 просмотров • 1 год назад

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

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

66,892 просмотров • 6 месяцев назад

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

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 просмотров • 5 месяцев назад

IN 1999 MIT FILMED A MATH LECTURE THAT QUIETLY BECAME THE FOUNDATION OF EVERY AI MODEL YOU'VE EVER USED AND ALMOST NO ONE WAS TAUGHT TO SEE IT THAT WAY 39 minutes from Gilbert Strang, who taught this at MIT for over 60 years -- the linear algebra course an entire generation of engineers and data scientists grew up on. -> The shift it creates: you stop seeing matrices as boring grids of numbers and start seeing them as the language of space, data, and motion itself. School drilled you to crunch matrices by hand and never told you why. Strang shows you what they actually mean. Every neural net, every embedding, every model you prompt is linear algebra running underneath. The math you skipped is the engine of the thing you use all day. Memorizing the steps was never the skill -> seeing what the numbers do is. This is where it finally clicks. Most people fear linear algebra and move on. The ones who watched this see straight into how AI actually works. Bookmark & Watch it today, this one's a legend ↓

IN 1999 MIT FILMED A MATH LECTURE THAT QUIETLY BECAME THE FOUNDATION OF EVERY AI MODEL YOU'VE EVER USED AND ALMOST NO ONE WAS TAUGHT TO SEE IT THAT WAY 39 minutes from Gilbert Strang, who taught this at MIT for over 60 years -- the linear algebra course an entire generation of engineers and data scientists grew up on. -> The shift it creates: you stop seeing matrices as boring grids of numbers and start seeing them as the language of space, data, and motion itself. School drilled you to crunch matrices by hand and never told you why. Strang shows you what they actually mean. Every neural net, every embedding, every model you prompt is linear algebra running underneath. The math you skipped is the engine of the thing you use all day. Memorizing the steps was never the skill -> seeing what the numbers do is. This is where it finally clicks. Most people fear linear algebra and move on. The ones who watched this see straight into how AI actually works. Bookmark & Watch it today, this one's a legend ↓

slash1s

433,431 просмотров • 1 месяц назад

Google just released a free 2-hour course on full Graph Engineering. How to go from one prompt to an agent graph that builds itself: 10:16 - Build your first AI agent 41:05 - Master prompt engineering 54:45 - Turn agents into graphs 1:20:10 - Run loops inside agent graphs 1:43:33 - Build a graph that builds itself Most people ship one agent and stop there. Google is teaching everything after: Prompt → Agents → Graphs → Loops → Self-Building Systems Single agents are the old workflow. Graphs that evolve themselves are the new one. This 2-hour watch is worth more than most paid agent engineering courses. Bookmark and watch it today Then read how to run 1,000 agents from one prompt below ↓

Google just released a free 2-hour course on full Graph Engineering. How to go from one prompt to an agent graph that builds itself: 10:16 - Build your first AI agent 41:05 - Master prompt engineering 54:45 - Turn agents into graphs 1:20:10 - Run loops inside agent graphs 1:43:33 - Build a graph that builds itself Most people ship one agent and stop there. Google is teaching everything after: Prompt → Agents → Graphs → Loops → Self-Building Systems Single agents are the old workflow. Graphs that evolve themselves are the new one. This 2-hour watch is worth more than most paid agent engineering courses. Bookmark and watch it today Then read how to run 1,000 agents from one prompt below ↓

Lunar

49,332 просмотров • 4 дней назад

ok starting to cook now you can now overlay new graphs on top of existing graphs to analyse the relations and flows between solana wallets visually see where they overlap, where the funds go etc all with a SINGLE Helius RPC endpoint

ok starting to cook now you can now overlay new graphs on top of existing graphs to analyse the relations and flows between solana wallets visually see where they overlap, where the funds go etc all with a SINGLE Helius RPC endpoint

mert

24,371 просмотров • 5 месяцев назад

Our new short course, Knowledge Graphs for RAG, is now available! Knowledge graphs are a data structure that is great at capturing complex relationships between data of multiple types. By enabling more sophisticated retrieval of text than similarity search alone, knowledge graphs can improve the context you pass to the LLM and the performance of your RAG applications. In this course, taught by Andreas Kollegger of Neo4j, you’ll - Explore how knowledge graphs work by building a graph of public financial documents from scratch - Learn to write queries that retrieve text and data from the graph and use it to enhance the context you pass to an LLM chatbot - Combine a knowledge graph with a question-answer chain to build better RAG-powered chat systems Sign up here!

Our new short course, Knowledge Graphs for RAG, is now available! Knowledge graphs are a data structure that is great at capturing complex relationships between data of multiple types. By enabling more sophisticated retrieval of text than similarity search alone, knowledge graphs can improve the context you pass to the LLM and the performance of your RAG applications. In this course, taught by Andreas Kollegger of Neo4j, you’ll - Explore how knowledge graphs work by building a graph of public financial documents from scratch - Learn to write queries that retrieve text and data from the graph and use it to enhance the context you pass to an LLM chatbot - Combine a knowledge graph with a question-answer chain to build better RAG-powered chat systems Sign up here!

Andrew Ng

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Are interactive essays the future of education? I asked Claude Code to build an interactive essay for PCA + SVD that works client side. Notice how python code executes and graphs are generated WITHOUT any backend! Play with it yourself 👇

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Paras Chopra

18,036 просмотров • 7 месяцев назад

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396,508 просмотров • 18 дней назад

Anthropic engineer at Stanford: “80% of our engineers are using self-improving loops. Now everyone is building agentic graphs.” in 1-hour Stanford lecture, Anthropic engineer reveals how Claude thinks - and why Graphs are the future. This 1-hour watch will replace your $500 course on agentic engineering. Watch it today, then read how to become a Graph Engineer in the article below.

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Codez

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Santiago

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INVASION 🚨🚨This is Miami, Florida, one of the largest led Republican cities. The Islamic Sharia Law is EVERYWHERE. The graphs show you how bad they are right now in Miami and surrounding areas. We're talking about 50 in Miami. This is unbelievable and it pains me to look where this is heading!

JOSH DUNLAP

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If you have time, watch the MIT lecture on Jacobian Matrices by Alan Edelman & Steven G. Johnson, & honestly, it may changes how you look at derivatives. Instead of treating derivatives as just formulas, they explain them as transformations of space. With proper rotation of images on live classes. haha Things they covered: - Difference between linear and non-linear transformations - How matrices act like operators that reshape geometry - Why the Jacobian is the best local linear approximation of a function - Understanding derivatives in higher dimensions - Intuition behind chain rule and automatic differentiation - Early foundations of backpropagation in ML One of the best parts was seeing how tiny input changes propagate through transformations mathematically. A really solid lecture if you want to build deeper intuition behind AI, optimization, and neural networks beyond just coding models. Full video link in comment.

If you have time, watch the MIT lecture on Jacobian Matrices by Alan Edelman & Steven G. Johnson, & honestly, it may changes how you look at derivatives. Instead of treating derivatives as just formulas, they explain them as transformations of space. With proper rotation of images on live classes. haha Things they covered: - Difference between linear and non-linear transformations - How matrices act like operators that reshape geometry - Why the Jacobian is the best local linear approximation of a function - Understanding derivatives in higher dimensions - Intuition behind chain rule and automatic differentiation - Early foundations of backpropagation in ML One of the best parts was seeing how tiny input changes propagate through transformations mathematically. A really solid lecture if you want to build deeper intuition behind AI, optimization, and neural networks beyond just coding models. Full video link in comment.

𝗿𝗮𝗺𝗮𝗸𝗿𝘂𝘀𝗵𝗻𝗮— 𝗲/𝗮𝗰𝗰

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Creator of Claude Code: “80% of our engineers are using self‑improving loops. Now everyone is building agentic Graphs" in 3-6 months, we’ll all be building graphs to orchestrate self‑improving agents. No more prompting.” In a 40-minute podcast, Boris Cherny reveals the future of agentic coding with Claude at AMD. Worth more than a $500 agentic course. Watch this video, then read the article below on how to become a graph architect.

Creator of Claude Code: “80% of our engineers are using self‑improving loops. Now everyone is building agentic Graphs" in 3-6 months, we’ll all be building graphs to orchestrate self‑improving agents. No more prompting.” In a 40-minute podcast, Boris Cherny reveals the future of agentic coding with Claude at AMD. Worth more than a $500 agentic course. Watch this video, then read the article below on how to become a graph architect.

Movez

158,748 просмотров • 14 дней назад

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Google just released a free 2-hour course On going from one prompt to a complete multi-agent graph: 17:44 - Build your first AI agent 39:30 - Create self-improving agent loops 1:12:38 - Connect those loops into graphs 1:34:26 - Make agents regulate themselves 1:55:05 - Orchestrate the full multi-agent system Most people are still building one agent and stopping there Google is teaching the entire stack: Prompt → Agents → Loops → Graphs → Multi-Agent Systems Single agents are the old workflow Self-regulating agent graphs are the new one This free 2-hour course is worth more than most paid agent engineering programs Bookmark and watch it today Then read the full architecture below ↓

Google just released a free 2-hour course On going from one prompt to a complete multi-agent graph: 17:44 - Build your first AI agent 39:30 - Create self-improving agent loops 1:12:38 - Connect those loops into graphs 1:34:26 - Make agents regulate themselves 1:55:05 - Orchestrate the full multi-agent system Most people are still building one agent and stopping there Google is teaching the entire stack: Prompt → Agents → Loops → Graphs → Multi-Agent Systems Single agents are the old workflow Self-regulating agent graphs are the new one This free 2-hour course is worth more than most paid agent engineering programs Bookmark and watch it today Then read the full architecture below ↓

rari

40,625 просмотров • 4 дней назад

Build better RAG by letting a team of agents extract and connect your reference materials into a knowledge graph. Our new short course, “Agentic Knowledge Graph Construction,” taught by Neo4j Innovation Lead Andreas Kollegger, shows you how. Knowledge graphs are an important way to store information accurately but they are a lot of work to build manually. In this course you’ll learn how to build a team of agents that turn data– in this case product reviews and invoices from suppliers–into structured graphs of entities and relationships for RAG. Learn how agents can automatically handle the time-consuming work of building graphs — extracting entities and relationships (e.g., Product "contains" Assembly, Part "supplied_by" Supplier, Customer review "mentions" Product), deduplicating them, fact-checking them, and committing them to a graph database — so your retrieval system can find right information to generate accurate output. For example, you can use agents to help trace customer complaints directly to specific suppliers, manufacturing processes, and product hierarchies, thus turning fragmented information into queryable business intelligence. Skills you’ll gain: - Build, store, and access knowledge graphs using the Neo4j graph database - Build multi-agent systems using Google’s Agent Development Kit (ADK) - Set up a loop of agentic workflows to propose and refine a graph schema through fact-checking - Connect agent-generated graphs of unstructured and structured data into a unified knowledge graph This course gets into the practicum of why knowledge graphs give more accurate information retrieval than vector search alone, especially for high-stakes applications where precision matters more than fuzzy similarity matching. Sign up here:

Build better RAG by letting a team of agents extract and connect your reference materials into a knowledge graph. Our new short course, “Agentic Knowledge Graph Construction,” taught by Neo4j Innovation Lead Andreas Kollegger, shows you how. Knowledge graphs are an important way to store information accurately but they are a lot of work to build manually. In this course you’ll learn how to build a team of agents that turn data– in this case product reviews and invoices from suppliers–into structured graphs of entities and relationships for RAG. Learn how agents can automatically handle the time-consuming work of building graphs — extracting entities and relationships (e.g., Product "contains" Assembly, Part "supplied_by" Supplier, Customer review "mentions" Product), deduplicating them, fact-checking them, and committing them to a graph database — so your retrieval system can find right information to generate accurate output. For example, you can use agents to help trace customer complaints directly to specific suppliers, manufacturing processes, and product hierarchies, thus turning fragmented information into queryable business intelligence. Skills you’ll gain: - Build, store, and access knowledge graphs using the Neo4j graph database - Build multi-agent systems using Google’s Agent Development Kit (ADK) - Set up a loop of agentic workflows to propose and refine a graph schema through fact-checking - Connect agent-generated graphs of unstructured and structured data into a unified knowledge graph This course gets into the practicum of why knowledge graphs give more accurate information retrieval than vector search alone, especially for high-stakes applications where precision matters more than fuzzy similarity matching. Sign up here:

Andrew Ng

168,153 просмотров • 11 месяцев назад

Dr. Robert Snellman's eight-hour course: The Fundamentals of Mathematics: Algebra, is available now. Enroll now for immediate access at Peterson Academy. Accompanying the course are practice questions as well! In this course, Dr. Snellman offers an accelerated and comprehensive introduction to algebra. He begins with basic variable manipulation and linear equations, then progresses to polynomials, quadratic equations, and exponential functions. Guided by Dr. Snellman, we build a strong mathematical foundation through a systematic study of functions, graphs, and solution methods, with a consistent emphasis on real-world applications in physics, finance, and cryptography. The course introduces increasingly sophisticated mathematical concepts and culminates in systems of linear equations and matrix operations, demonstrating how algebraic thinking can model and solve meaningful, practical problems.

Dr. Robert Snellman's eight-hour course: The Fundamentals of Mathematics: Algebra, is available now. Enroll now for immediate access at Peterson Academy. Accompanying the course are practice questions as well! In this course, Dr. Snellman offers an accelerated and comprehensive introduction to algebra. He begins with basic variable manipulation and linear equations, then progresses to polynomials, quadratic equations, and exponential functions. Guided by Dr. Snellman, we build a strong mathematical foundation through a systematic study of functions, graphs, and solution methods, with a consistent emphasis on real-world applications in physics, finance, and cryptography. The course introduces increasingly sophisticated mathematical concepts and culminates in systems of linear equations and matrix operations, demonstrating how algebraic thinking can model and solve meaningful, practical problems.

Peterson Academy

59,142 просмотров • 1 год назад

Still the best hour on graph engineering ever recorded, Andrew Ng breaking down how to build agentic knowledge graphs from scratch: 00:00 - what agentic knowledge graphs actually are 03:05 - building a graph from scratch 13:58 - the architecture behind multi-agent systems 22:57 - building a real one with Google ADK 01:06:02 - why graphs are the future of AI agents I've seen $500 courses that teach less. Watch it, then take it further with my step-by-step guide on graph engineering below.

Still the best hour on graph engineering ever recorded, Andrew Ng breaking down how to build agentic knowledge graphs from scratch: 00:00 - what agentic knowledge graphs actually are 03:05 - building a graph from scratch 13:58 - the architecture behind multi-agent systems 22:57 - building a real one with Google ADK 01:06:02 - why graphs are the future of AI agents I've seen $500 courses that teach less. Watch it, then take it further with my step-by-step guide on graph engineering below.

Anatoli Kopadze

261,639 просмотров • 13 дней назад