Geometric Bites

Understanding Eigenvectors and Eigenvalues: A Geometric Perspective Every linear transformation has a hidden structure. Most vectors are pushed into new directions when a matrix acts on them, but a handful of special vectors behave differently. These are the eigenvectors — directions that remain perfectly aligned with themselves, even after the transformation has been applied. To understand how a matrix works, you must understand these special directions. What Is an Eigenvector? An eigenvector of a matrix A is a non-zero vector x that satisfies the relation A x = λ x The number λ is the eigenvalue associated with x . This equation expresses a simple but striking fact: the transformation does not rotate the vector at all. The direction is preserved exactly. The only change is a scaling by the factor λ . A positive eigenvalue stretches the vector. A value between 0 and 1 compresses it. A negative eigenvalue reverses the direction. But in every case, the vector re...