Eigenvalues eigenvectors

• How to Compute the Semisimple Part of a Matrix
  Compute a matrix’s semisimple part by replacing each Jordan block with its eigenvalue times the identity, or by evaluating a suitable polynomial.
• Eigenvalues of the Left-Shift Plus Right-Shift Operator
  For the unilateral shifts on ℓ² of the nonnegative integers, their sum has spectrum [−2, 2] but no eigenvalues.
• How to Diagonalize a 3×3 Matrix
  To diagonalize a 3×3 matrix, find its eigenvalues and eigenvectors, check that three independent eigenvectors form P, and verify AP = PD.
• How to Find a Basis for Each Eigenspace
  Find an eigenspace by solving (A − λI)x = 0. A basis consists of independent eigenvectors that span the solutions, excluding zero.