Eigenvalues and Eigenvectors of Real Matrices

Exercise 1 : Eigenvalues — 2x2 real matrix

For $A = \begin{pmatrix} 4 & 2 \ 1 & 3 \end{pmatrix}$ find the eigenvalues.

Exercise 2 : Real vs Complex Eigenvalues — 2x2 question

Is it possible for a real $2 \times 2$ matrix to have one real and one complex eigenvalue?

Exercise 3 : No Real Eigenvalues — rotation-like matrix

Prove that there are no real eigenvalues of the matrix $A = \begin{pmatrix} 3 & 2 \ -2 & 1 \end{pmatrix}$

Exercise 4 : Eigenvalues and Eigenvectors — 3x3 computation

  1. For $A = \begin{pmatrix} -1 & -1 & 1 \ -4 & 2 & 4 \ -1 & 1 & 5 \end{pmatrix}$ find all the eigenvalues.
  2. Find corresponding eigenvectors.

Exercise 5 : Defective Matrix — repeated eigenvalue

Find the roots of the characteristic equation for $A = \begin{pmatrix} 1 & 1 \ 0 & 1 \end{pmatrix}$. Then try to find corresponding eigenvectors.

Exercise 6 : Diagonalization Criteria — theory and practice

  1. State the necessary and sufficient conditions for a real matrix to be diagonalizable over $\mathbb{R}$.
  2. Decide whether the matrix $B=\begin{pmatrix} 2 & 0 & 0 \ 0 & 3 & 1 \ 0 & 0 & 3 \end{pmatrix}$ is diagonalizable. Justify your answer.

Exercise 7 : Algebraic and Geometric Multiplicity — multiplicities

For $C=\begin{pmatrix} 4 & 1 & 0 \ 0 & 4 & 1 \ 0 & 0 & 4 \end{pmatrix}$ compute the algebraic multiplicity and geometric multiplicity of the eigenvalue $4$. Explain the relationship between them and consequences for diagonalizability.

Exercise 8 : Symmetric Matrices — orthogonal diagonalization

  1. Show that a real symmetric matrix has real eigenvalues and orthogonal eigenvectors (state the spectral theorem for symmetric matrices).
  2. Compute the eigenvalues and an orthonormal set of eigenvectors for $D=\begin{pmatrix} 2 & -1 \ -1 & 2 \end{pmatrix}$.

Exercise 9 : Complex Eigenpairs and Real Matrices — conjugate pairs

Let $E=\begin{pmatrix} 0 & -1 \ 1 & 0 \end{pmatrix}$.

  1. Compute the eigenvalues and eigenvectors of $E$ over $\mathbb{C}$.
  2. Explain why complex eigenvalues of real matrices occur in conjugate pairs and how to form real 2D invariant subspaces from such pairs.

Exercise 10 : Jordan Blocks and Generalized Eigenvectors — small Jordan example

  1. For $F=\begin{pmatrix} 5 & 1 \ 0 & 5 \end{pmatrix}$ find the Jordan normal form and a chain of generalized eigenvectors.
  2. Explain why the presence of a nontrivial Jordan block prevents diagonalization.

Exercise 11 : Matrix Exponential via Diagonalization and Jordan Form

  1. If $A$ is diagonalizable with $A=SDS^{-1}$, express $e^{tA}$ in terms of $S,D$.
  2. Compute $e^{tF}$ for $F$ from Exercise 10 using its Jordan form.

Exercise 12 : Applications — mechanical system and modes

Consider the matrix $M=\begin{pmatrix} 0 & 1 \ -2 & -3 \end{pmatrix}$ arising from a simple damped oscillator linearization.

  1. Find eigenvalues and determine whether the system is overdamped, underdamped, or critically damped.
  2. If eigenvectors exist, write the general solution of $\dot{x}=Mx$ using eigen-decomposition (or explain the Jordan approach if defective).