Diagonalization is a powerful technique for simplifying matrix operations. It allows us to represent a matrix as a product of simpler matrices, making calculations easier and revealing important properties of linear transformations.

This topic builds on our understanding of eigenvalues and eigenvectors, showing how they can be used to break down complex matrices. We'll explore the conditions for diagonalizability and learn how to construct diagonal matrices, unlocking new ways to solve problems in linear algebra.

Diagonalizability of matrices

Conditions for diagonalizability

• Matrix A diagonalizable if and only if it has n linearly independent eigenvectors (n dimension of matrix)
• Algebraic multiplicity of eigenvalue counts occurrences as root of characteristic polynomial
• Geometric multiplicity of eigenvalue measures dimension of associated eigenspace
• Diagonalizability requires geometric multiplicity equal algebraic multiplicity for each distinct eigenvalue
• Matrices with n distinct eigenvalues guaranteed diagonalizable
• Symmetric matrices always diagonalizable regardless of eigenvalue multiplicity

Testing for diagonalizability

• Compare sum of dimensions of all eigenspaces to matrix dimension
• Analyze characteristic polynomial roots and corresponding eigenspaces
• Check for linear independence of eigenvectors
• Examine special cases (symmetric matrices, distinct eigenvalues)
• Calculate algebraic and geometric multiplicities for each eigenvalue
• Verify if sum of geometric multiplicities equals matrix dimension

Diagonalization process

Constructing diagonal and change of basis matrices

• Form diagonal matrix D by placing eigenvalues along main diagonal (repeat according to algebraic multiplicity)
• Build change of basis matrix P using eigenvectors as columns (correspond to respective eigenvalues in D)
• Diagonalization equation expressed as $A = PDP^{-1}$ (P^(-1) inverse of P)
• Columns of P form eigenbasis for vector space
• Maintain consistent order between eigenvectors in P and eigenvalues in D
• Find linearly independent eigenvectors for repeated eigenvalues to complete P

Steps for diagonalization

• Solve characteristic equation $det(A - λI) = 0$ to find eigenvalues
• Compute eigenvectors for each eigenvalue using $(A - λI)v = 0$
• Organize eigenvectors into change of basis matrix P
• Create diagonal matrix D with eigenvalues on main diagonal
• Verify diagonalization by calculating $PDP^{-1}$ and comparing to original matrix A
• Handle cases with repeated eigenvalues by finding generalized eigenvectors if necessary

Applications of diagonalization

Solving systems of differential equations

• Simplify solutions for systems $dx/dt = Ax$ (A constant coefficient matrix)
• General solution given by $x(t) = Pe^{Dt}c$ (c vector of constants from initial conditions)
• Compute matrix exponential $e^{Dt}$ by exponentiating individual diagonal entries
• Transform coupled system into decoupled system for easier solving
• Determine stability by examining eigenvalues in D
• Complex eigenvalues introduce oscillatory behavior (trigonometric functions)
• Long-term system behavior governed by eigenvalue with largest real part

Other applications

• Power method for finding dominant eigenvalue and eigenvector
• Solve recurrence relations and difference equations
• Analyze Markov chains and steady-state distributions
• Optimize quadratic forms in multivariable calculus
• Implement principal component analysis in data science
• Model vibration modes in mechanical systems

Diagonalization and eigenvalues vs eigenvectors

Relationship between diagonalization and eigenstructure

• Eigenvalues of A become diagonal entries of D (scaling factors in eigendirections)
• Eigenvectors of A form columns of P (directions where A acts as scalar multiple)
• Algebraic and geometric multiplicities determine diagonalizability and P construction
• Eigendecomposition of A written as $A = \sum_{i=1}^n λ_iP_i$ (λᵢ eigenvalues, Pᵢ projection matrices onto eigenspaces)
• Characteristic polynomial $det(A - λI) = 0$ yields eigenvalues for eigenvector calculation
• Eigenvectors of A^n same as A (eigenvalues raised to nth power)

Properties and theorems

• Trace of A equals sum of eigenvalues
• Determinant of A equals product of eigenvalues
• Similar matrices share same eigenvalues (different eigenvectors)
• Algebraic multiplicity always greater than or equal to geometric multiplicity
• Sum of algebraic multiplicities equals matrix dimension
• Eigenvalues of triangular matrices appear on main diagonal
• Real symmetric matrices have real eigenvalues and orthogonal eigenvectors