Eigenvectors & Eigenvalues: Invariant Axes of Linear Transformations
Visualize how matrices shear space while keeping characteristic eigenvector directions on their own span.
For a square matrix A in R^{n x n}, a non-zero vector v is an eigenvector of A with corresponding eigenvalue lambda if applying the linear transformation A to v merely scales v by the scalar factor lambda without rotating it off its span: A v = lambda v.
Most vectors in 2D or 3D space get knocked off their original line when a matrix transformation stretches and shears the coordinate grid. Eigenvectors are the special 'axis-of-stretch' directions that remain locked on their original line through the origin—merely stretched, compressed, or flipped by a factor of lambda.
Eigenvectors & Eigenvalues (EigenTransformationScene)
Step-by-Step Symbolic LaTeX Derivation
Follow each algebraic transformation from first principles to computational implementation.
Rewrite Scaling Condition as Homogeneous System
Multiplying v by scalar lambda is identical to multiplying v by the scaled identity matrix lambda * I.
Non-Trivial Kernel Requires Zero Determinant
A matrix sends a non-zero vector to the origin if and only if the transformation squashes volume to zero.
Diagonalization & Fast Matrix Powers
In the eigenbasis P, applying A k times reduces to raising each scalar eigenvalue to the k-th power.
If a 2x2 rotation matrix rotates every vector in R^2 by 90 degrees counterclockwise, what are its real eigenvalues?
What does the Spectral Theorem guarantee for real symmetric matrices?
Every real symmetric matrix A = A^T has exclusively real eigenvalues and a complete set of mutually orthogonal eigenvectors (A = Q Lambda Q^T).