Algebraic vs Geometric Multiplicity of Eigenvalues! Let's determine when a matrix is diagonalizable!

When we find eigenvalues of a matrix, we often discover that some of them appear more than once. But did you know there are two ways to count their occurrence? That's where geometric and algebraic multiplicity come in!

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You may wonder what’s the difference between geometric and algebraic multiplicity in eigenvalues, and why it matters for diagonalization. In this video, we break it down with clear explanations and visuals!

🔹 Algebraic multiplicity: The number of times an eigenvalue appears as a root of the characteristic equation.

🔹 Geometric multiplicity: The number of linearly independent eigenvectors associated with an eigenvalue (i.e., the dimension of its eigenspace).

Why does this matter? A matrix is diagonalizable if and only if its geometric multiplicity equals its algebraic multiplicity for every eigenvalue. If the geometric multiplicity is smaller, the matrix cannot be fully diagonalized!

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This video is a math lecture about eigenvalues, eigenvectors, algebraic multiplicity and geometric multiplicity as part of my math series about linear algebra, matrices and machine learning. Subscribe to my channel if you want to learn more about math and cool math tricks!

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