29.2.1. Steps to Find Eigenvalues

Try these first

1. 1.

   What is the first step to find eigenvalues of a square matrix?

   Hint

   Remember, it must be a square matrix!

2. 2.

   Define eigenvalue in your own words.

   Hint

   Think about the relationship between matrices and their transformations.

3. 3.

   What is the formula used to find eigenvalues from a matrix?

   • \\(det(A) = 0\\)
   • \\(det(A - \\lambda I) = 0\\)
   • \\(A\\cdot v = \\lambda v\\)

   Hint

   Think about eigenvectors' relationship with eigenvalues.

4. 4.

   True or False: All matrices have eigenvalues.

   • True
   • False

   Hint

   Consider the properties of square matrices.

5. 5.

   Given the matrix $D = \begin{bmatrix} 1 & 2 & 3 \\ 0 & 4 & 5 \\ 0 & 0 & 6 \end{bmatrix}$, find its eigenvalues and explain your process.

   Hint

   What property of triangular matrices simplifies finding eigenvalues?

6. 6.

   Translate the significance of eigenvalues in the context of finding principal stresses in civil engineering materials.

   Hint

   Consider structural integrity and design safety.