What you can now do.

• §6.1 Eigenvectors. The rubber sheet: when a matrix moves the plane, most arrows turn, but a few only get longer, shorter or flipped. Those are the eigenvectors, and the stretch factor is the eigenvalue: Ax = λx. Any multiple of an eigenvector works too, so we look for lines.
• §6.1 Finding them. The radio dial: turn λ until A − λI squashes a line flat, which means det(A − λI) = 0. Solve that for λ, then use elimination on A − λI to get each eigenvector. Check with the trace (sum of λs) and the determinant (product of λs). A triangular matrix shows its eigenvalues on its diagonal.
• §6.2 Diagonalizing. The slanted streets: count in eigenvector steps, and A just multiplies each count by its λ. Eigenvectors go in X, eigenvalues in Λ, in the same order: A = XΛX⁻¹. Powers are easy, Aᵏ = XΛᵏX⁻¹. A shear has too few eigenvectors and can't be diagonalized.
• §6.2 Powers and stability. The two jars and the two cities: every step multiplies each eigenvector part by its λ. Parts with |λ| < 1 die out, |λ| > 1 grow, λ = 1 stays. A Markov matrix always has λ = 1, and its eigenvector, scaled to add to 1, is the steady state as shares. For head counts, multiply by the total (the same total as u₀).
• §6.4 Symmetric matrices. The pizza dough pulled two ways: real eigenvalues and perpendicular eigenvectors. Unit eigenvectors make an orthogonal Q, so A = QΛQᵀ, a sum of stretched projections λqqᵀ.
• §6.5 Positive definite. The marble in a bowl: all eigenvalues positive, xᵀAx > 0 for every nonzero x, all pivots positive. One negative eigenvalue makes a saddle.

Next lesson: the singular value decomposition (Strang chapter 7)

Eigenvectors only work for square matrices, and they may not be perpendicular. The SVD fixes both: every matrix, even a rectangular one, turns a circle into an ellipse along perpendicular axes. It does for every matrix what A = QΛQᵀ does for symmetric ones, and it leads to image compression and PCA.

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