One Page Note for AI

# Example: N = d = d_v = 4, scale = 1/sqrt(d) = 1/2 Q = [ [1, 0, 2, 0],   [0, 1, 1, 0],   [1, 1, 0, 1],   [0, 2, 1, 1] ] K = [ [2, 1, 0, 0],   [0, 1, 1, 0],   [1, 0, 1, 1],   [0, 0, 2, 1] ] V = [ [1, 0],   [0, 1],   [1, 1],   [2, 1] ] ======================== 1) NORMAL ATTENTION ======================== Scores: S = (Q K^T) / 2  →  S ∈ R^{4×4} S = [ [1.0, 1.0, 1.5, 2.0],   [0.5, 1.0, 0.5, 1.0],   [1.5, 0.5, 1.0, 0.5],   [1.0, 1.5, 1.0, 1.5] ] Row-wise softmax P = softmax_row(S)  (approximate decimals) P ≈ [ [0.1570, 0.1570, 0.2588, 0.4269],   [0.1887, 0.3113, 0.1887, 0.3113],   [0.4271, 0.1571, 0.2589, 0.1571],   [0.1888, 0.3112, 0.1888, 0.3112] ] Output: O = P V  (approximate) O ≈ [ [1.2696, 0.8427],   [1.0000, 0.8113],   [1.0000, 0.5731],   [1.0000, 0.8112] ] ======================================== 2) FLASHATTENTION TILING (same numbers) =======================================...

 1. Eigen decomposition for a square matrix:    Av = λv:      an eigenvector v is a direction that A only scales      (by eigenvalue λ) without rotation.    To find eigenvalues λ, solve det(A - λI) = 0.    Then express A = VΛV⁻¹:      eigen decomposition rewrites A in terms of its eigenvectors V      and eigenvalues Λ. 2. Singular Value Decomposition (SVD) for any matrix:    Eigen decomposition of XᵀX:      eigenvectors = right singular vectors V,      eigenvalues  = squared singular values σ².    Eigen decomposition of XXᵀ:      eigenvectors = left singular vectors U,      eigenvalues  = squared singular values σ².    Full factorization:      X = UΣVᵀ:        U (m×m) = left singular vectors,        V (n×n) = right singular vectors,   ...