Eigenvalues and eigenvectors

Eigenvectors are unchanged in direction by a linear transformation

Eigenvectors remain unchanged in direction when a linear transformation is applied. This unique property allows them to be scaled by a constant factor, known as the eigenvalue. Understanding eigenvectors and eigenvalues is crucial in various applications, including solving systems of linear equations and analyzing stability in dynamical systems.

Example

Consider a linear transformation represented by the matrix A = [[2, 0], [0, 3]]. The eigenvector v = [1, 0] remains unchanged in direction when transformed by A, as A * v = [2, 0] = 2 * v. The corresponding eigenvalue λ = 2 indicates that v is scaled by a factor of 2.

Recognizing eigenvectors and eigenvalues helps in simplifying complex problems in linear algebra and understanding the behavior of systems over time.

Related concepts

Principal component analysis

Eigenvectors point along maximum variance

ill-conditioned matrices cause numerical instability: small input changes → large output changes

Ill-conditioned matrices amplify input perturbations, leading to significant output variability

Ordinary least squares

OLS minimizes squared differences

The elastic net combines L1 and L2: λ₁|w| + λ₂w² gives both sparsity and stability

Elastic net: λ₁|w| + λ₂w² enforces sparsity and stability simultaneously

Cholesky decomposition

Cholesky decomposition factors A = LL^T for symmetric positive definite matrices

the determinant tells you about volume scaling under a linear transformation

The determinant of a matrix representing a linear transformation indicates the factor by which volumes are scaled

One email a day: 5 concepts + the 5 stories that matter →

Swipe through 100 ML concepts daily

Open TickerNews