Discover the intrinsic structure of matrices through eigenanalysis.
An eigenvector of A is a nonzero vector whose direction is preserved (or reversed) under the transformation A — only its length scales by the factor λ. This captures the "natural axes" of the transformation.
A = [[3, 1], [0, 2]]
det(A − λI) = (3 − λ)(2 − λ) − 0 = λ² − 5λ + 6 = 0
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For λ = 3: (A − 3I)v = 0 → v₁ = (1, 0). For λ = 2: v₂ = (−1, 1).
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For symmetric matrices, the Spectral Theorem guarantees real eigenvalues and orthogonal eigenvectors: A = QDQᵀ.