29.7. Special Case: Symmetric Matrices

Try these first

1. 1.

   What defines a symmetric matrix?

   Hint

   Remember the fundamental property that relates symmetric matrices to their transposes.

2. 2.

   True or False: All eigenvalues of symmetric matrices are complex.

   Hint

   Recall the specific characteristics of symmetric matrices.

3. 3.

   What property do all eigenvalues of symmetric matrices have?

   • They are all complex
   • They are all real
   • They can be either real or complex

   Hint

   Think about the characteristics of symmetric matrices we just discussed.

4. 4.

   True or False: Eigenvectors of symmetric matrices corresponding to distinct eigenvalues are always orthogonal.

   • True
   • False

   Hint

   Relate this to the inner product properties of vectors.

5. 5.

   Given a symmetric matrix A, prove that all of its eigenvalues are real.

   Hint

   Consider applying the spectral theorem.

6. 6.

   In an engineering application, a stress tensor is symmetric and its eigenvalues represent principal stresses. If the principal stresses are found to be λ1 = 5, λ2 = 3, λ3 = 2, how would you interpret these in terms of material selection?

   Hint

   Think about the implications of stress distribution in materials.