EM 214 Diagonalization

Diagonalizable matrix

• A_nxn is diagonalizable if there is an invertible matrix P such that P^-1AP is a diagonal matrix.

• A_nxn is diagonalizable if A has n linearly independent eigenvectors.

Find P & D from P^-1AP = D

• Find eigenvectors of A.

• Matrix of eigenvectors as columns is P and is invertible.

  • Eigenvectors only need to be LI.

• Use eigenvalues of A as diagonal of diagonal matrix D.

• P and D's columns must be lined up AKA eigenvalue and relevant eigenvector have to be in same column position.

dim(E_λk) and being diagonalizable

• Matrix A_nxn is diagonalizable only if dim(E_λ1) + … + dim(E_λk) = n

• Sum of eigenspace dimensions must be equal to matrix size n.

Orthogonally diagonalizable

• Matrix A_nxn is orthogonally diagonalizable if P in P^-1AP = D is orthogonal.

• A is orthogonally diagonalizable only if A is symmetric.

Matrix (A_nxn)^k

• To raise a matrix A_nxn to the power k:

  • Use A^k = PD^kP^-1. Do D^k = [a^k, b^k, …] = R first, then do PRP^-1.

Similar matrices

• A and B are similar if P^-1AP = B.

Similarity equivalence rules

• det(A) = det(B)

• A and B have same characteristic equation, therefore same eigenvalues.

• tr(A) = tr(B)

P^-1AP and PAP^-1

• If we can find a matrix P such that P^-1AP = D, then we can find a matrix P such that PAP^-1.

• If we can find an orthogonal matrix P such that P^TAP = D, then we can find an orthogonal matrix P such that PAP^T.

Find orthogonal matrix P such that P^-1AP = D

• Same process as finding non-orthogonal P, but eigenvectors used to form P need to be an orthonormal set.

• AKA normalise eigenvectors before using in P.

Find matrix from given eigenvalues & corresponding eigenvectors

• P is matrix of eigenvectors as columns in order.

• D is diagonal matrix with eigenvalues as diagonal in order.

• Use P^-1AP = D → A = PDP^-1 to find A.

• Use EROs to find P^-1.