: Eigenvectors define the principal axes of data variance, allowing for dimensionality reduction in machine learning.
(A−λI)v=0open paren cap A minus lambda cap I close paren bold v equals 0 must be non-zero, the matrix must be singular, meaning its determinant is zero: Eigenvalues and Eigenvectors
A=(4123)cap A equals the 2 by 2 matrix; Row 1: 4, 1; Row 2: 2, 3 end-matrix; : : Eigenvectors define the principal axes of data
det(A−λI)=0det of open paren cap A minus lambda cap I close paren equals 0 This polynomial equation in is called the . 3. Geometric Interpretation A linear transformation 4. Practical Example Consider the matrix
typically moves vectors in various directions. However, eigenvectors are special directions where the transformation only results in scaling (stretching or shrinking) rather than rotation. The eigenvalue represents the scale factor. 4. Practical Example Consider the matrix