Algebraic vs Geometric Multiplicity of Eigenvalues! What Happens if Geometric is less than Algebraic

Описание: 🎓 What happens when geometric multiplicity of an eigenvalue is less than algebraic multiplicity of an eigenvalue?
In this continuation of our deep dive into eigenvalues, we explore a key case:
➡️ When algebraic multiplicity = 2
➡️ But geometric multiplicity = 1

This signals a defective eigenvalue, meaning the matrix is not diagonalizable.

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➡️   • Vectors + Linear Algebra + Matrices + Mach...  

Key Moments
00:50 Algebraic Multiplicity Definition
02:20 Geometric Multiplicity Definition
04:15 Example) Determine the Algebraic and Geometric Multiplicity of the Eigenvalues of Matrix A
05:28 Finding Eigenvalues (Algebraic Multiplicity)
07:05 Matrix Determinant Formula
10:20 Finding Eigenvectors (Geometric Multiplicity)
17:05 Is the Matrix Diagonalizable?

🧠 In This Video, You'll Learn:
✅ What is algebraic multiplicity?
– The number of times an eigenvalue appears as a root of the characteristic polynomial.

✅ What is geometric multiplicity?
– The number of linearly independent eigenvectors associated with an eigenvalue (i.e., the dimension of the eigenspace).

✅ Why this matters:
– If geometric multiplicity is less than algebraic multiplicity, then the matrix is not diagonalizable and is called defective. This is a crucial concept in advanced linear algebra and is especially relevant in differential equations, control theory, and machine learning algorithms like PCA.

🔍 This Video Includes:
📌 Clear definitions of algebraic and geometric multiplicity
📌 Explanation of the relationship:
    geometric multiplicity less than or equal to algebraic multiplicity
📌 Step-by-step example where:
    - Algebraic multiplicity = 2
    - Geometric multiplicity = 1
📌 Proof that the matrix is not diagonalizable
📌 How to determine if a matrix is defective
📌 Important matrix operations:
  – Characteristic polynomial
  – Eigenvalue solving
  – Determinants
  – Matrix multiplication

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📺 Watch the Full Series:
🎥 [Part 1] Algebraic vs Geometric Multiplicity (Intro & Diagonalizable Matrices) :    • Algebraic vs Geometric Multiplicity of Eig...  
🎥 Eigenvalues & Eigenvectors Crash Course:    • Find Eigenvalues and Eigenvectors FAST! St...  
🎥 Matrix Diagonalization & Eigendecomposition:    • What is Eigen Decomposition and Matrix Dia...  
🎥 PCA & ML Applications of Eigenvalues:    • Eigenvalues Crash Course | Step-by-Step Gu...  

Be sure to check out the full playlist on Linear Algebra & ML Foundations to deepen your understanding.

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This video is a math lecture about determining the algebraic and geometric multiplicity of eigenvalues and understanding whether a matrix is diagonalizable or not as part of my math series about linear algebra. Subscribe to my channel if you want to learn more about math and cool math tricks!

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