Eigendecomposition Fundamentals

Linear Algebra for Machine Learning · Axiom Academy

Full decomposition of a 3×3 matrix Perform complete eigendecomposition of this symmetric 3×3 matrix: Step 2: Find Orthonormal Eigenvectors Step 3: Form Orthogonal Matrix Q Step 4: Form Diagonal Matrix D Step 5: Eigenvalue Decomposition The original matrix A can be understood as a sum of three rank-1 matrices: 5 times the projection onto the first eigenvector direction Plus 2 times the projection onto the second eigenvector direction Plus 1 time the projection onto the third eigenvector direction This decomposition shows that the "action" of the matrix is a weighted sum of stretches in the eigenvector directions. The eigenvalues tell you how much to stretch in each direction!

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