What are Eigenvalues and Eigenvectors?
In linear algebra, an **eigenvector** of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding **eigenvalue** is the factor by which the eigenvector is scaled.
Mathematically, for a given square matrix **A**, a non-zero vector **v** is an eigenvector if it satisfies the equation:
Av = λv
Here, **v** is the eigenvector, **A** is the matrix, and **λ** (lambda) is the scalar eigenvalue. This means that when the matrix **A** acts on the vector **v**, the resulting vector is parallel to **v**, simply scaled by the value **λ**. Eigenvectors and eigenvalues have important applications in fields like physics (e.g., vibration analysis), computer graphics, and machine learning (e.g., Principal Component Analysis).
