diagonalization

Summary

Diagonalizable

ie. the eigenvectors have to span

Eigenspaces are linearly independent

Equivalent statements for diagonalizability

Scalar matrices

Concept

Diagonalization

several linearly independent eigenvectors can share the same eigenvalue

Geometric and algebraic multiplicities

• the geometric multiplicity cannot be greater than the algebraic multiplicity, in order for the matrix to be diagonalizable

if the algebraic is 1, then the geometric is also 1 -> only need to check geometric for eigenvalues with algebraic > 1

Non-diagonalizable

• show that the characteristic polynomial doesn’t split into linear factors
• show that there is a geometric multiplicity that is less than its algebraic multiplicity

Powers of diagonalizable matrices

akin to changing base, applying the power, then changing back

Application

Diagonal and identity matrix

any diagonal matrix is diagonalizable by the identity matrix

Non-diagonalizable

Extra

Theorem for the independence of eigenspaces(formally)

Octave

# Diagonalization
[P D] = eig(A)