Spectral theorem for symmetric matrices
General · Mathematics
Study notes
Q: Spectral-decompose [[2,1],[1,2]]. Symmetric ✓. Eigen: 3 ((1,1)/√2), 1 ((1,-1)/√2). Q = (1/√2)[[1,1],[1,-1]] (orthogonal!). D = diag(3,1). A = QDQᵀ ✓. Check: QDQᵀ = (1/2)[[3+1,3-1],[3-1,3+1]] = [[2,1],[1,2]] ✓! (Orthonormal eigenbasis!)