29.11. QR Algorithm for Eigenvalue Computation

Try these first

1. 1.

   What does the QR algorithm compute?

   Hint

   Think about what eigenvalues tell us about transformations.

2. 2.

   What are the two components of QR decomposition?

   Hint

   One is orthogonal, the other has zeros below the diagonal.

3. 3.

   What two matrices does the QR algorithm decompose a matrix into?

   • Upper triangular and diagonal
   • Orthogonal and lower triangular
   • Orthogonal and upper triangular

   Hint

   Recall the properties of Q and R.

4. 4.

   True or False: The QR algorithm can only be used for symmetric matrices.

   • True
   • False

   Hint

   Consider the general applicability in engineering.

5. 5.

   Given a symmetric matrix, explain why the QR algorithm would likely converge faster than for a non-symmetric matrix.

   Hint

   Focus on the properties of symmetry and how they relate to matrix behavior.

6. 6.

   Illustrate a civil engineering scenario that requires the application of the QR algorithm, detailing the benefits it offers.

   Hint

   Consider the dynamic response of structures to external forces.