Theory of eigensystems-Review

Every nxn real matrix has n real eigenvalues counting multiplicities.

• True

• False

A matrix A is unitarily diagonalizable if and only if ...

• A has a full set of linearly independent eigenvectors.

• A has disctinct eigenvalues.

• A is normal.

What is the minimal polynomial of the nxn identity matrix?

• λ-1

• (λ-1)ⁿ

• λⁿ

• λ

Let the Jordan canonical form of A have p Jordan blocks. Which of the following statements is correct?

• A has p linearly independent eigenvectors.

• A has p distinct eigenvalues.

• The eigenvalue λ has algebraic multiplicity p.

Let μ be an eigenvalue of A that belongs to exactly two 3x3 Jordan blocks. Select the correct statement.

• The eigenvalue μ has algebraic multiplicity 3 and geometric multiplicity 2.

• The eigenvalue μ has geometric multiplicity 2 and index 3.

• The eigenvalue μ has algebraic multiplicity 6 and index 2.

Let p be a scalar polynomial and let λ be an eigenvalue of A.

Select the correct statement.

• λ is an eigenvalue of p(A).

• p(λ) be an eigenvalue of p(A).