The directions a matrix does not turn: how to find them, A = XΛX⁻¹ and powers, stability and steady states, symmetric matrices and positive definite bowls. 7 short slides, then worked examples and problems. Each slide has an everyday picture, a worked example and Python you can run right on the page.
Free preview: most of every slide is yours to read, with its Python lab. Members get the full text, every example with its solution, and five extra problems per lesson.
Lesson 7EigenvectorsEigenvectors: the arrows a matrix only stretchesOpen slide →01 · Strang §6.1 · Introduction to eigenvaluesAn eigenvector is a direction that a matrix only stretches, never turns: Ax = λx. Find one by dragging an arrow, with examples and Python. Strang §6.1.
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Lesson 7det(A − λI) = 0Finding eigenvalues: solve det(A − λI) = 0Open slide →02 · Strang §6.1 · The characteristic equationFind eigenvalues by solving det(A − λI) = 0, then eigenvectors from the null space of A − λI. Trace and determinant checks, examples and Python. Strang §6.1.
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Lesson 7A = XΛX⁻¹Diagonalizing a matrix: A = XΛX⁻¹Open slide →03 · Strang §6.2 · Diagonalizing a matrixWith enough eigenvectors, A = XΛX⁻¹: eigenvectors in X, eigenvalues on the diagonal of Λ. Then powers are easy, Aᵏ = XΛᵏX⁻¹. Examples and Python. Strang §6.2.
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Lesson 7StabilityPowers of A: which parts grow, shrink or stayOpen slide →04 · Strang §6.2 · Powers, stability and Markov matricesRepeating A makes parts grow or decay by their eigenvalues. Markov matrices settle to a steady state, the eigenvector for λ = 1. Python examples. Strang §6.2.
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Lesson 7Symmetric matricesSymmetric matrices: perpendicular eigenvectorsOpen slide →05 · Strang §6.4 · Symmetric matricesA symmetric matrix has real eigenvalues and perpendicular eigenvectors, so A = QΛQᵀ with Q orthogonal: the spectral theorem. Examples and Python. Strang §6.4.
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Lesson 7Positive definitePositive definite matrices: every eigenvalue positiveOpen slide →06 · Strang §6.5 · Positive definite matricesA symmetric matrix is positive definite when all its eigenvalues are positive, or xᵀAx > 0 for x ≠ 0: a bowl, not a saddle. Python examples. Strang §6.5.
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Lesson 7RecapWhat you can now do.Open slide →07 · Recap · ReadingChapter 6 on one page: Ax = λx, det(A − λI) = 0, A = XΛX⁻¹, powers and stability, Markov steady states, symmetric and positive definite matrices.
