Skip to content
Free public beta until December 4, 2026: every course is open, no card needed. After that, the first chapter of each course stays free and full access is a one-time $39 ($19 for up to 100 founding members chosen for helpful feedback). Your progress is kept. How the beta works

linear-algebra

Eigenvalues and the characteristic equation

taught

Taught in

Requires

Required by

Local knowledge map

Local knowledge map

Circulant matrices are diagonalized by the DFT basis Classifying critical points via the Hessian (bowls, domes,… Classifying nodes Deflation Diagonalizing to canonical form Eigenspaces From boundary conditions to a Sturm-Liouville… Fundamental matrices Invariant subspaces and upper-triangular form Minimal versus characteristic polynomial Orthogonality of Sturm-Liouville eigenfunctions Polynomial division, roots, and factorization Repeated eigenvalues and generalized eigenvectors Self-adjoint operators and real eigenvalues Singular Sturm-Liouville problems, previewed Solving the heat equation on a finite rod with Robin… Sturm-Liouville form The Fast Fourier Transform: divide-and-conquer an… The Fourier transform in several dimensions, and separability The Rayleigh quotient The eigenvalue method -- complex eigenvalues The inverse DFT and conjugate symmetry for real signals The inverse power method The separation-of-variables ansatz Eigenvalue problems from boundary conditions Every operator on a complex space has an eigenvalue Hyperbolicity: real, distinct eigenvalues Multivariable first- and second-order optimality conditions The DFT, defined and as a matrix transformation The characteristic polynomial The eigenvalue method -- distinct real eigenvalues The power method for eigenvalues The regular Sturm-Liouville problem Eigenvalues and the characteristic equation Aliasing Boundary-value problems, contrasted with IVPs Complex arithmetic for linear algebra The gradient in three variables The linear differential operator L The shooting method Vector-valued solutions and the vector Wronskian Writing a first-order system in matrix form The eigenvector question Regularization geometry: what ridge does to the normal…

Click to expand

Solid arrows are required prerequisites; dashed are recommended or helpful. Lower rows are more foundational. Click any box to jump to that concept.