31.15. Similarity over Complex Field

Try these first

1. 1.

   What does it mean for a matrix to be similar to another matrix?

   Hint

   Think about the definition involving the invertible matrix.

2. 2.

   Can a matrix with complex eigenvalues be diagonalizable?

   Hint

   Consider the implications of complex numbers.

3. 3.

   What indicates that a matrix can be diagonalized over the complex field?

   • It has only real eigenvalues
   • It has complex eigenvalues
   • It cannot be diagonalized

   Hint

   Think about the role of eigenvalues in diagonalization.

4. 4.

   True or False: A matrix with eigenvalues ±i can be diagonalized over real numbers.

   • True
   • False

   Hint

   Remember the properties of complex eigenvalues.

5. 5.

   Prove whether the matrix A = $\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}$ is diagonalizable over the complex field. Show your work.

   Hint

   Calculate the determinant of A - λI.

6. 6.

   Discuss the implications of using only real eigenvalues for modeling vibrations in structures during a seismic event. What could result from this oversight?

   Hint

   Think about the nature of vibrations and how they affect structural integrity.