Characteristic Polynomial

Linear Algebra for Machine Learning · Axiom Academy

Finding eigenvalues through the characteristic equation Deriving the Characteristic Equation Starting with our fundamental eigenvalue equation: This means (A - λI) multiplied by v gives zero. Since v is non-zero, the matrix (A - λI) must be singular (not invertible). A matrix is singular if and only if its determinant equals zero: This is the characteristic equation ! When you expand det(A - λI), you get a polynomial in λ. This is the characteristic polynomial . The eigenvalues are the roots of the characteristic polynomial. That is, the values of λ that make the characteristic polynomial equal to zero. Set the determinant equal to zero and solve for λ The solutions are your eigenvalues For a 2×2 matrix, you can use the quadratic formula. For larger matrices, you may need numerical methods. Algebraic and Geometric Multiplicity Algebraic Multiplicity: The number of times an eigenvalue appears as a root of the characteristic polynomial. Geometric Multiplicity: The dimension of the eigenspace (the number of linearly independent eigenvectors) for that eigenvalue. Example: The polynomial λ² - 4λ + 4 = (λ - 2)² has eigenvalue λ = 2 with algebraic multiplicity 2. But it might have geometric multiplicity 1 or 2 depending on the original matrix.

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