Eigenvalues and Eigenvectors 2

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date: 2025-01-23

Let be an eigenvalue of that is is a root of the char polynomial and A is a matrix of wrt some basis

Algebraic multiplicity of is the highest power of that divides

Geometric multiplicity of is the dimension of eigenspace (this is a subspace of )

Theorem

Let be an eigenvalue of . then geometric multiplicity algebraic multiplicity of

Let be the gm. of let be an ordered basis of Extend the basis to basis Matrix of wrt is of the form

This shows that divides so

Exercise

If a matrix is a triangular matrix then the diagonal elements are the eigenvalues

Definition

A linear operator is said to be diagonalizable if a basis of consisting of eigenvectors of

is said to be diagonalizable if A is similar to a diagonal matrix

Suppose is diagonalizable let be distinct eigenvalues of and wrt this basis matrix of is a diagonal matrix, the diagonal entries being

Theorem

If are distinct eigenvalues of and if is an eigenvector corresponding to then is a linearly independent set of vector

Proposition

if are the eigenspace corresponding to the distinct eigenvalues of , say then is direct sum

Proposition

If is a diagonalizable operator and if are the distinct eigencalues of then