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- SVD by Hand: Splitting a 2×3 Matrix into UΣVᵀSingular value decomposition breaks any matrix into a rotation, a stretch and another rotation. Working one 2×3 matrix all the way through: eigenvalues of AAᵀ, square roots for the singular values, U and V column by column, and a check that multiplies it all back. Along the way, why the square root and why signs can go wrong.