Applied Mathematical Methods

9

Diagonalization and Similarity Transformations

Diagonalization of matrices is the focus of the present chapter. First, we examine the conditions that guarantee diagonalizability through similarity transformations. Next, if a matrix is not diagonalizable, we explore other canonical forms that facilitate the identification of eigenvalues and eigenvectors. Finally, we study the special properties of symmetric matrices in relation to the eigenvalue problem.

Diagonalizability

Consider a matrix A ∈ R^n×n having n linearly independent eigenvectors v₁, v₂, ⋯, v_n with corresponding eigenvalues λ₁, λ₂, ⋯, λ_n. Assembling these vectors into a matrix S = [v₁ V₂ ⋯ v_n], we note that

where Λ is the diagonal matrix containing the eigenvalues as the diagonal ...