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
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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)=PeDtc (c vector of constants from initial conditions)
Compute matrix exponential eDt by exponentiating individual diagonal entries
Transform coupled system into decoupled system for easier solving