Linear Algebra: Diagonalization

Linear Algebra - Week 6 - Diagonalization

Learning Objectives

• Students will be able to calculate matrix eigenvalues.
• Students will be able to calculate matrix eigenvectors.
• Students will be able to determine if a matrix is diagonalizable.
• Students will be able to diagonalize a given matrix.

Exercises

• Find the eigenvalues for given matrices, and find a basis for the corresponding eigenspace of each eigenvalue for matrices A, B, C, D, E, and F.
• Diagonalize matrices A, B, C, and D if possible.
• Matrix A is not diagonalizable.
• A matrix A is invertible if and only if 0 is not an eigenvalue of A.
• If A is an n x n diagonalizable matrix $A^2$ is diagonalizable.
• If A and B are n x n diagonalizable matrices, A + B is diagonalizable if and only if AB = BA.

Suggested Questions

• What is an eigenvalue?
• What is an eigenvector?
• How do you calculate the eigenvalues of a matrix?
• How do you calculate the eigenvectors of a matrix?
• What is a diagonalizable matrix?
• How do you diagonalize a given matrix?
• Why diagonalize a matrix?

Statics

Chapter 3: Equilibrium

• Equilibrium of a particle

Equilibrium of a Particle

• A particle is in equilibrium if the resultant force acting on it is zero.
• The equation for equilibrium is $\sum \vec{F} = 0$.
• In component form, the equilibrium equations are $\sum F_x = 0$, $\sum F_y = 0$, and $\sum F_z = 0$.
• Use a Free-Body Diagram (FBD) to show the particle and all forces acting on it.

Example 1

• A 200-kg crate is supported by several ropes and pulleys.
• Determine the force in rope A.

Solution

• Draw the FBD at D, showing: $F_{A}$ (Force in rope A), $F_{DB}$ (Force in rope DB), and the angles.
• Equations of Equilibrium:
  • $\sum F_x = 0: F_{DB} \cos 30^\circ - F_A = 0$
  • $\sum F_y = 0: F_{DB} \sin 30^\circ - w = 0$
  • Where $W = mg$
• $F_{DB} = \frac{W}{\sin 30^\circ} = \frac{200 \cdot 9.81}{\sin 30^\circ} = 3924 N$
• $F_A = F_{DB} \cos 30^\circ = 3924 \cdot \cos 30^\circ = 3398 N$

Springs

• The Force Exerted by a Spring: $F = ks$
  • F = Force
  • k = Spring constant (stiffness)
  • s = Displacement (elongation or compression)

Cables and Pulleys

• Cables have negligible weight, cannot stretch, and have the same tension throughout.
• Pulleys are frictionless and have the same cable tension on both sides.

Coplanar Force Systems

• Equilibrium equations simplify to:
  • $\sum F_x = 0$
  • $\sum F_y = 0$

3-D Force Systems

• Equilibrium equations are:
  • $\sum F_x = 0$
  • $\sum F_y = 0$
  • $\sum F_z = 0$

Learn to calculate eigenvalues and eigenvectors. Determine if a matrix is diagonalizable and diagonalize matrices when possible. Explore the relationship between invertibility, eigenvalues, and diagonalizability.