eigen values

for linear algebra

How are the eigenvalues of A and A² related? Are they ever the same?

• For any matrix A: If λ is an eigenvalue of A, then λ² is an eigenvalue of A².

• So, the eigenvalues of A² are the squares of the eigenvalues of A, not generally the same.

• In case of identity matrix, they are same.

What is the relationship between the eigenvalues and eigenvectors of AB and BA for matrices A (m×n) and B (n×m)?

AB and BA have the same nonzero eigenvalues (with the same multiplicities), but their eigenvectors and the number of zero eigenvalues can differ.
If ABx=λx, then BA(Bx)=λ(Bx)BA, so eigenvectors are related but not the same.

When is a matrix diagonalizable in terms of eigenvalues and eigenvectors?

A matrix is diagonalizable if it has n linearly independent eigenvectors (for an n×n matrix).
This often happens when it has n distinct eigenvalues, but repeated eigenvalues can also work if their geometric multiplicity equals their algebraic multiplicity.

A real 3×3 matrix with all eigenvalues nonzero and all eigenvectors linearly dependent must be diagonalizable and invertible.

False!

• Nonzero eigenvalues ⇒ matrix is invertible.

• But if all eigenvectors are linearly dependent, the matrix is not diagonalizable (needs 3 independent eigenvectors).

• So, invertible but not diagonalizable.