Unit 10 of 10

Matrix Functions and Applications

  • 5 lessons
  • Videos 51 to 55

This unit computes powers and exponentials of matrices, then closes the course with two capstone problems and an exit workshop.

Jordan chains explain growth that eigenvalues alone do not predict, and the matrix exponential is built from a convergent series without assuming differential equations. The capstones split a coupled update into independent modes and compare pseudoinverse, truncated and regularized reconstructions from noisy data. The last lesson asks you to work unfamiliar proofs and calculations before seeing the solutions.

Every lesson will be one video. The videos are coming soon, and each lesson page already has the definitions and an example for every key term.

Matrix Powers and Exponentials

  1. Matrix Powers and Polynomial Functions

    How to compute powers and polynomials of a matrix by reduction modulo the minimal polynomial and the binomial formula for a Jordan block, and how Jordan chains add polynomial factors to the powers.

    Video 51

  2. The Matrix Exponential

    Why the exponential series converges for every square matrix, how the matrix exponential behaves under similarity and for commuting sums, how to compute it for a Jordan block, and what transient growth is.

    Video 52

Capstones in Structure and Reconstruction

  1. Coupled Systems and Eigenmodes

    How a linear update with a symmetric coupling matrix splits into eigenmodes, using mean and difference coordinates, the mode decomposition and an energy quadratic form, and what changes when the coupling is nonsymmetric.

    Video 53

  2. Inverse Problems and Regularization

    How small singular values amplify noise in an inverse problem, how truncated SVD and Tikhonov regularization control it through filter factors, the residual and solution norm tradeoff, and what no method can recover.

    Video 54

Exit Workshop

  1. Proofs and Problem Solving

    Worked methods for the dual of a quotient space, the normality test, Jordan basis verification, orthogonal projectors as self-adjoint idempotents, the forward difference operator and method choice under measurement noise.

    Video 55