Eigenbasis

Proposition: Let be two eigenvectors of , , and be two associated eigenvectors Then, are linearly independent. Proof: Consider for now,

Proof by induction: By induction on For : TBA

Algebraic and geometric multiplicity

• How many eigen values can a matrix have?
  • Let
    • This is a polynomial with real entries
  • The fundamental theorem of algebra has complex zeros counted with multiplicity
• An eigenvalue of can be a multiple zero of the characteristic polynomial
  • That multiplicity of is called the algebraic multiplicity of
• Recall: The eigenspace of is the subspace of all eigenvectors of
  • The dimension of this subspace is called the geometric multiplicity of

Def. of Eigenbasis

Definition: Eigenbasis. A basis of is called an eigenbasis of if the basis vectors are eigenvectors of

• What are the properties of an eigenbasis?
  • It can be used for diagonalization

Main result 12/2/2025

For the following are equivalent

1. is diagonalizable
2. Algebraic multiplicity is equal to the geometric multiplicity for all eigenvalues of

Proof: Let , an eigenvalue Then, geometric multiplicity of alg. multiplicity of
HW: , where is , is , is , and is . Then, Proof cont. Suppose the eigenspace associated to has dimension . Let be a bass of this eigenspace. Now, complete to a bass of by considering a basis of . Say the basis is . Note that . Define and consider Since Coming back to the proof: Since , alg. mult. of

• Recall:
  • This shows that Similar matrices have the same characteristic polynomial

Proof: Case 1: be an eigenbasis of associated to . Define, Consider Which is a diagonal matrix, so this case is proven. Case 2: Since is diagonalizable, there exists an invertible such that is a diagonal matrix
Therefore, are eigenvectors associated with eigenvalue . Thus, is an eigenbasis of associated to Case 3 Alg. mult. = geo. mult. for all eigenvalues sum of geo. mult. = sum of all dim of eigenspaces = There exists an eigenbasis Case 4: There exists an eigenbasis. This implies that the sum of dimension of eigenspace is equal to the sum of geo. mult. which is less that the sum of alg. mult. Therefore, geo. mult. is equal to the alg. mult. for each eigenvalue.

Corollary: has -distinct eigenvalues, then is diagonalizable.

Corollary: Let be a polynomial. If is diagonalizable, then so is . Proof: There exists non-singular such that is Substituting this into our polynomial Therefore, So, is similar to the diagonal matrix Hence, it is also diagonalizable

Spectral theorem: Recall: is called symmetric if is called Hermitian if Let denote the dot product. , . Therefore,

Proposition: Let is such that . Then, All eigenvalues of are real numbers Proof: Let be an eigenvector of with an associated eigenvector Now, Therefore, Therefore, , meaning that is real.

Proposition: Let and be eigenvectors of associated with with with . Then, Proof:

Definition: A matrix is said to have simple spectrum if all its eigenvalues are different. Lemma: Any symmetric matrix can be approximated by matrices with simple spectrum.